Pre-Calc 11 · Lesson 3

Geometric Sequences.

A geometric sequence is a number ladder where every rung is the previous one times the same factor, so growth and decay compound multiplicatively instead of stacking by a fixed step.

By the end, you can

1I can decide whether a sequence is geometric by checking if the ratio of consecutive terms is constant, and state the common ratio r.
2I can write and use the general term tn = t1 * rⁿ⁻¹ to find any term, the first term, or the common ratio.
3I can solve for the number of terms n in a geometric sequence using roots and logarithms.
4I can set up and solve a two-equation system when I am given two non-adjacent terms to find t1 and r.
5I can model real situations of growth and decay (doubling cells, fading colour, bouncing balls, populations) with geometric sequences and interpret the results.

01 / CONCEPTS

The core ideas.

What makes a sequence geometric: the common ratio

A sequence is geometric when each term after the first is found by multiplying the previous term by the same non-zero constant r, called the common ratio. You test for this by dividing any term by the one before it. If every such quotient is the same, the sequence is geometric. For 2, 4, 8, 16, ... each division gives 2, so r = 2.

r = \dfrac{t_n}{t_{n-1}}

Why it works

Arithmetic sequences ask 'what do I ADD each step?' Geometric sequences ask 'what do I MULTIPLY by each step?' Dividing consecutive terms cancels everything except that one repeated multiplier, which is exactly why the ratio exposes r. Because r is multiplied (not added), the gaps between terms grow or shrink over time instead of staying constant.

The general term tn = t1 * rⁿ⁻¹

To reach the nth term you start at t1 and multiply by r repeatedly. Getting to t2 takes one multiplication, t3 takes two, and tn takes (n-1) multiplications. So tn = t1 times r raised to the power (n-1). Knowing any three of t1, r, n, and tn lets you solve for the fourth.

t_n = t_1\, r^{\,n-1}

Why it works

Write the terms vertically: t1 = t1, t2 = t1*r, t3 = t1*r*r, t4 = t1*r³. The exponent on r is always one less than the term number because the first term has had zero multiplications applied. The exponent literally counts how many times you have multiplied by r since the start, which is why it is n minus 1, not n.

Finding r when terms are not adjacent

If you know two terms that are several steps apart, the jump between them is a power of r. To go from t3 to t6 you multiply by r three times, so t6 = t3 * r³. Divide the two terms to isolate r^(gap), then take the matching root. The gap in exponents equals the gap in term positions.

\dfrac{t_m}{t_k} = r^{\,m-k}

Why it works

Each step forward multiplies by exactly one r, so moving from position k to position m multiplies by r exactly (m - k) times. The ratio of the two terms isolates that bundled r^(m-k); the root then unbundles it. Watch the sign: an odd root of a negative number is negative, so a negative ratio is possible and real.

Growth versus decay: what r tells you

The size of r controls the behaviour. If r > 1 the terms grow (a population, doubling bacteria). If 0 < r < 1 the terms shrink toward zero (a fading colour, a bouncing ball losing height). A negative r makes the terms alternate sign while changing size. A percent change of p% means r = 1 + p/100 for growth or r = 1 - p/100 for decay.

r = 1 \pm \dfrac{p}{100}

Why it works

Losing 8% per day does NOT mean r = 0.08. After losing 8% you KEEP 92%, so r = 0.92. The classic mistake is multiplying by the percent lost instead of the percent remaining. Always ask: after one step, what fraction of the previous amount is left? That fraction is r.

Solving for n with roots and logarithms

When the unknown is the position n, the variable sits in the exponent. First isolate rⁿ⁻¹ by dividing both sides by t1. If the right side is a clean power of r you can read off the exponent. Otherwise take the logarithm of both sides, which brings the exponent down to ground level so you can solve for n.

n - 1 = \dfrac{\log\!\left(t_n/t_1\right)}{\log r}

Why it works

A logarithm is the inverse of exponentiation: it answers 'r to WHAT power gives this number?' Since geometric growth hides the answer inside an exponent, the log is the only tool that pulls that exponent out into the open where ordinary algebra can finish the job.

02 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

A bacteria sample starts with 10 cells and each doubles: 10, 20, 40, 80, 160, ... Find t1 and r, and write the general term.

Identify the first term

The sequence starts at t1 = 10.

Find the common ratio

Divide consecutive terms: 20/10 = 2, 40/20 = 2, 80/40 = 2. So r = 2.

Write the general term

tn = t1 * rⁿ⁻¹ = 10(2)ⁿ⁻¹.

EX 2easy

A photocopier reduces a 25 cm photo to 67% of its size each pass. What is the length after 5 reductions, to the nearest tenth of a cm?

Set up the values

t1 = 25, r = 0.67. The original is term 1, so after 5 reductions you want term 6: n = 6.

Apply the general term

t6 = 25(0.67)⁶⁻¹ = 25(0.67)⁵.

Compute

(0.67)⁵ = 0.13501..., so t6 = 25 * 0.13501 = 3.375...

State the answer

About 3.4 cm.

EX 3medium

A geometric sequence has 3, 12, 48, 5y + 7, ... Find the value of y.

Find r

12/3 = 4 and 48/12 = 4, so r = 4.

Find the 4th term

t4 = 48 * 4 = 192.

Set the expression equal

5y + 7 = 192.

Solve for y

5y = 185, so y = 37.

EX 4medium

In a geometric sequence the 3rd term is 54 and the 6th term is -1458. Find t1 and r, and list the first three terms.

Relate the two terms

From t3 to t6 is three steps, so t6 = t3 * r³: -1458 = 54 r³.

Solve for r

r³ = -1458/54 = -27, so r = cube root of -27 = -3. (Negative is allowed since this is an odd root.)

Find t1

t3 = t1 * r²: 54 = t1(-3)² = 9 t1, so t1 = 6.

List the first three terms

6, 6(-3) = -18, -18(-3) = 54. They match t3 = 54.

EX 5hard

A piano's lowest note A0 is 27.5 Hz and the highest note C8 is 4186.009 Hz. The 88 keys form a geometric sequence. Find the common ratio from key 1 to key 88, then verify using keys 1 to 4 (C1 = 32.7 Hz).

Set up the full-keyboard equation

t1 = 27.5, n = 88, tn = 4186.009. Then 4186.009 = 27.5 * r⁸⁸⁻¹ = 27.5 r⁸⁷.

Isolate the power

r⁸⁷ = 4186.009 / 27.5 = 152.218...

Take the 87th root

r = 152.218^(1/87) = 1.0594...

Verify with keys 1 to 4

32.7 = 27.5 r³, so r³ = 32.7/27.5 = 1.1891, and r = cube root = 1.0594...

Conclude

Both give r is about 1.06, confirming one consistent ratio across the keyboard.

EX 6hard

A bouncing ball is dropped from 3.0 m and rebounds to 75% of its previous height each bounce. After how many bounces does it rise to about 40 cm (0.40 m)?

Set up the model

Let the heights form a geometric sequence with the 3.0 m drop as t1 = 3.0 and r = 0.75. Then t2 is the height after the 1st bounce, t3 after the 2nd, and in general tₖ₊₁ is the height after the kth bounce. We want a term equal to 0.40.

Write the equation

0.40 = 3.0(0.75)ⁿ⁻¹.

Isolate the power

(0.75)ⁿ⁻¹ = 0.40/3.0 = 0.13333...

Take logs

n - 1 = log(0.13333)/log(0.75) = 7.00..., so n - 1 = 7 and n = 8.

Interpret

The 8th term (t8) is the height after the 7th bounce. Check: after bounce 6 the height is 3.0(0.75)⁶ = 0.534 m and after bounce 7 it is 3.0(0.75)⁷ = 0.400 m. So the ball first reaches about 40 cm after the 7th bounce.

EX 7challenge

If x + 2, 2x + 1, and 4x - 3 are three consecutive terms of a geometric sequence, find x, the common ratio, and the three terms.

Use the constant-ratio property

In a geometric sequence the middle term squared equals the product of its neighbours: (2x + 1)² = (x + 2)(4x - 3).

Expand both sides

Left: 4x² + 4x + 1. Right: 4x² + 5x - 6.

Solve

4x + 1 = 5x - 6, so x = 7.

Find the terms

x + 2 = 9, 2x + 1 = 15, 4x - 3 = 25. So the terms are 9, 15, 25.

Find r

r = 15/9 = 5/3, and check 25/15 = 5/3. Confirmed.

03 / COMMON TRAPS

Where students slip.

Trap

Treating a decay rate as the common ratio: if something loses 8% each day, students use r = 0.08.

Fix

After losing 8% you keep 92% of the previous amount, so r = 0.92. Use r = 1 - 0.08. Only the surviving fraction multiplies forward. (This is exactly why Mala, not Alex, is correct in the aquarium problem.)

Trap

Confusing geometric with arithmetic and adding a constant instead of multiplying, or treating a constant percent loss as a constant subtraction.

Fix

Check the ratio, not the difference. In 40, 36.8, 33.86, ... the differences (-3.2, -2.94, ...) are NOT equal, but the ratios (0.92, 0.92, ...) ARE. A constant percent change is multiplicative, so it is geometric, not arithmetic.

Trap

Using the wrong exponent, writing tn = t1 * rⁿ instead of t1 * rⁿ⁻¹.

Fix

The first term has been multiplied by r zero times, so its exponent is 0. The nth term needs (n-1) multiplications. Always subtract one: t5 uses r⁴, not r⁵. A quick check: plug n = 1 and you must get t1, which only works with rⁿ⁻¹.

Trap

Assuming a negative ratio is impossible because you cannot take an even root of a negative, so r must be positive.

Fix

Geometric ratios can be negative. When the term gap is odd (like t3 to t6, a gap of 3), you take an odd root, and the cube root of -27 is a real -3. A sequence like 6, -18, 54, -108 is perfectly geometric with r = -3.

04 / QUIZ

Test yourself.

The Test

32 problems across three tiers. Auto-graded on submit. Hints available before you submit, full solutions after.

0/32Answered

Core

Core (12)

Which sequence is geometric?

Need a hint?

Divide each term by the one before it. Are the quotients all equal?
For 1, 2, 4, 8: 2/1 = 2, 4/2 = 2, 8/4 = 2.

For the sequence 3, -9, 27, -81, ..., state the common ratio r and the general term tn.

Need a hint?

Divide -9 by 3.
-9/3 = -3, and 27/-9 = -3, so r = -3.

Find a formula for the nth term when r = 2 and t1 = 3.

Need a hint?

Plug directly into tn = t1 rⁿ⁻¹.

Find a formula for the nth term of 192, -48, 12, -3, ...

Need a hint?

Divide -48 by 192 to get r.
-48/192 = -1/4.

Determine the first four terms of the geometric sequence with t1 = 4 and r = -3.

Need a hint?

Multiply repeatedly by -3.

Determine the first four terms of the geometric sequence with t1 = 2 and r = 0.5.

Need a hint?

Halve each term.

A geometric sequence has t1 = 5, r = 3, and a term tn = 135. Find n.

Need a hint?

Set 135 = 5(3)ⁿ⁻¹ and isolate 3ⁿ⁻¹.
3ⁿ⁻¹ = 27 = 3³, so n - 1 = 3.

A geometric sequence has t1 = -2, r = -3, and tn = -1458. Find n.

Need a hint?

Divide: (-1458)/(-2) = 729.
(-3)ⁿ⁻¹ = 729 = 3⁶, and the exponent must be even to keep the sign positive.

In a geometric sequence t1 = 8.1 and t5 = 240.1. Find the common ratio r (positive).

Need a hint?

From t1 to t5 is four steps: t5 = t1 r⁴.
r⁴ = 240.1/8.1 = 29.64..., then take the 4th root.

C10

Fill the blanks for 6, 18, 54, ___, ___ . The common ratio is r = ___.

Need a hint?

18/6 = 3.

C11

For the sequence 1.28, 0.64, 0.32, ..., what is the common ratio?

Need a hint?

Divide 0.64 by 1.28.

C12

A student claims 10, 15, 22.5, 33.75, ... is geometric with r = 1.5. Verify whether this is correct.

Need a hint?

Check 15/10 and 22.5/15.

Apply

Apply (14)

In a geometric sequence t3 = 5 and t6 = 135. Find r and t1, then write tn.

Need a hint?

t6 = t3 r³.
r³ = 135/5 = 27, so r = 3; then t3 = t1 r².

A geometric sequence has t1 = 4 and t13 = 16384. Find r (positive) and the formula tn.

Need a hint?

t13 = t1 r¹².
r¹² = 16384/4 = 4096; take the 12th root.

A pair of jeans fades by 5% with each wash, starting at 100% colour. (a) What is r? (b) What percent of colour remains after 10 washes? (c) How many washes until only 25% remains?

Need a hint?

Fading 5% means keeping 95%, so r = 0.95.
After 10 washes use 100(0.95)¹⁰; for 25% solve 0.95ⁿ = 0.25 with logs.

A ball is dropped from 3.0 m and rebounds to 75% of its previous height each bounce. (a) Write t1 and r. (b) What height does it reach after the 6th bounce, to the nearest cm?

Need a hint?

The drop height 3.0 is t1; each rebounce multiplies by 0.75.
After the 6th bounce you want the 7th term: t7 = 3.0(0.75)⁶.

Pincher Creek wind turbines generated 326 MW in 2004 and a projected 10000 MW by 2010. Modelling this as a geometric sequence (one term per year, 6 steps), find the annual growth rate.

Need a hint?

From 2004 to 2010 is 6 yearly steps: 10000 = 326 r⁶.
r⁶ = 10000/326; take the 6th root.

In One Grain of Rice, Rani receives 1 grain on day 1, doubled each day. (a) Write the general term. (b) How many grains on day 30?

Need a hint?

t1 = 1, r = 2.
Day 30 means t30 = 2²⁹.

Frog Georges jumped 191.41 cm, then 197.34 cm, then 203.46 cm, approximating a geometric sequence over 5 jumps. (a) Find the ratio r to three decimals. (b) How long was his winning 5th jump, to the nearest tenth of a cm?

Need a hint?

r = 197.34/191.41.
The 5th jump is t5 = 191.41 r⁴.

Jason trains by jumping 2 sledges in week 1, 4 in week 2, 8 in week 3, doubling weekly. In how many weeks does he first reach at least 142 sledges?

Need a hint?

Week n means 2ⁿ sledges.
2⁷ = 128 (too few), 2⁸ = 256.

A boat swing ride travels 96% as far on each successive swing. The ride ends when the arc length is 30 m, which happens on the 20th swing. Find the arc length of the first swing, to the nearest tenth of a metre.

Need a hint?

t20 = 30, r = 0.96, n = 20.
t1 = t20 / r¹⁹ = 30 / 0.96¹⁹.

A10

Kidneys filter out 18% of a medicine every 2 hours, starting from 250 mL. (a) How much remains after 12 h, to the nearest tenth of a mL? (b) After how many hours does it drop below 20 mL?

Need a hint?

Keeping 82% means r = 0.82; each period is 2 h, so 12 h = 6 periods.
(a) 250(0.82)⁶. (b) Solve 250(0.82)^k < 20 for the number of 2 h periods k.

A11

A coiled basket has a first coil radius of 6 mm and a consecutive-coil ratio of 1.22. Find the radius of the 8th coil, to the nearest tenth of a mm.

Need a hint?

t8 = 6(1.22)⁷.

A12

The Arctic Winter Games are held every 2 years. In 1970 there were 700 competitors and in 2008 there were 2000. Modelling growth geometrically (one term per games), find the rate of growth per games, to the nearest tenth of a percent.

Need a hint?

From 1970 to 2008 is 38 years, which is 19 games intervals.
2000 = 700 r¹⁹; take the 19th root.

A13

Alex, Mala, and Paul model an aquarium that holds 40 L and loses 8% of its water to evaporation each day, asking for the water at the start of day 7. Alex uses r = 0.08, Mala uses r = 0.92, Paul subtracts 3.2 L each day. Which student is correct and why?

Need a hint?

After losing 8%, what fraction of water remains?
Losing 8% keeps 92%, so r = 0.92, not 0.08. The loss is a percent (multiplicative), not a fixed amount (so not arithmetic).

A14

Yeast cells start from a single cell and double each period. (a) Write the general term. (b) How many cells after 25 doublings?

Need a hint?

t1 = 1, r = 2.
After 25 doublings is the 26th term: 2²⁵.

Extension

Extension (6)

In a set of 50 Russian nesting dolls the tallest is 60 cm and the smallest is 1 cm, with sizes forming a geometric sequence. Find the common ratio to three decimal places.

Need a hint?

t1 = 60, t50 = 1, so t50 = t1 r⁴⁹.
r⁴⁹ = 1/60; take the 49th root.

On a guitar, the nut-to-bridge distance is 38 cm, first-fret-to-bridge is 35.87 cm, and second-fret-to-bridge is 33.86 cm, approximating a geometric sequence. Find the distance from the 8th fret to the bridge, to the nearest tenth of a cm.

Need a hint?

r = 35.87/38.
The nut is t1 = 38; the 8th fret distance is the 9th term: t9 = 38 r⁸.

Frog Georges (first jump 191.41 cm, ratio 1.031) wants to beat Santjie's record of 10.2 m (1020 cm). If he kept improving geometrically, how many jumps would he need to first exceed 1020 cm?

Need a hint?

Solve 191.41(1.031)ⁿ⁻¹ = 1020 for n.
(1.031)ⁿ⁻¹ = 1020/191.41 = 5.328; take logs to get n - 1 = 54.8.

Prove that if a, b, c, ... is an arithmetic sequence, then 6^a, 6^b, 6^c, ... is a geometric sequence.

Need a hint?

Arithmetic means b - a = c - b = d (a constant).
Look at the ratio of consecutive new terms, 6^b / 6^a, using the law 6^b/6^a = 6^(b-a).

If x + 2, 2x + 1, and 4x - 3 are three consecutive terms of a geometric sequence, find x, the three terms, and the common ratio.

Need a hint?

In a geometric sequence, the middle term squared equals the product of its neighbours.
(2x + 1)² = (x + 2)(4x - 3); expand and solve.

A car battery's charge follows C = 100(0.98)^d as a percent, where d is days. (a) Rewrite this as a geometric sequence general term tn. (b) On about which day does the charge first drop below 50%?

Need a hint?

The formula C = 100(0.98)^d uses d as the exponent; a sequence term uses n - 1.
Solve 100(0.98)^d = 50, so 0.98^d = 0.5 with logs.