Geometric sequences: worked examples and method

The common ratio

Divide any term by the one before it. If the answer is always the same, that number is the common ratio r.

Using the ratio

Multiply by r to go forward to the next term, and divide by r to go back to the previous term.

The nth term

The nth term is a x r^(n-1). The first term has no r in it, so the power is one less than the position.

Check n = 1

Put n = 1 into your formula. It must give the first term. If it does not, the power is wrong.

Terms far apart

If two terms are k places apart, the later term divided by the earlier term is r^k. Take the kth root to find r.

Percentages

Growth of p% has ratio 1 + p/100 and decay of p% has ratio 1 - p/100. Year 0 is the first term, so year n is term n + 1.

Spotting a geometric sequence and finding the ratio

To test whether a sequence is geometric, divide each term by the one before it. If the results are all the same, the sequence is geometric and that number is the common ratio. The sequence 2, 6, 18, 54 gives 6 ÷ 2 = 3, 18 ÷ 6 = 3 and 54 ÷ 18 = 3, so r = 3. The sequence 4, 7, 10, 13 gives 7 ÷ 4 and 10 ÷ 7, which are different, so it is not geometric. It is arithmetic, because the differences are all 3. Always check at least three terms before you decide.

The common ratio can be a fraction or negative. For 80, 20, 5, 1.25 the ratio is 20 ÷ 80 = 1/4, so each term is a quarter of the one before. For 2, -6, 18, -54 the ratio is -6 ÷ 2 = -3, and the signs alternate. A ratio between 0 and 1 shrinks the sequence and a ratio bigger than 1 grows it. To find the next term, multiply the last one by r. To find the previous term, divide by r.

Method: 1. Divide the second term by the first, then the third by the second, and compare. 2. If the answers match, write the common ratio r. If not, check whether the differences match instead. 3. Use r to continue the sequence forwards by multiplying, or backwards by dividing.

Tip: Arithmetic sequences share a difference. Geometric sequences share a ratio. Test both before deciding.

The nth term, a r to the n minus 1

The first term is a. The second is a r, the third is a r^2, and the fourth is a r^3. Each step multiplies by one more r, but the first term has no r at all, so the power is always one less than the position. The nth term is u_n = a r^n-1. For 5, 10, 20, 40 we have a = 5 and r = 2, so u_n = 5 × 2^n-1. Check the formula with n = 1: 5 × 2^0 = 5, the first term. If the check fails, the power is wrong.

To find a particular term, substitute its position. The 8th term of 5, 10, 20, 40, dots is 5 × 2^7 = 5 × 128 = 640. Calculate the power first and then multiply by a, because the power applies to r only. Writing (5 × 2)^7 is a very large number and wrong. Notice how fast the terms grow. The nth term is nothing like the linear an + b form of an arithmetic sequence: a power of n in the exponent makes the terms change by a multiplying factor each time.

Method: 1. Write down a (the first term) and r (the common ratio). 2. Put them into u_n = a r^n-1 and check it gives the first term when n = 1. 3. Substitute the position you need, work out the power first, then multiply by a.

Tip: Check n = 1 gives the first term. That one test catches a wrong power.

Finding unknown terms and unknown ratios

You may be given two terms that are not next to each other. Suppose the 2nd term is 12 and the 5th term is 324. Going from the 2nd to the 5th is 3 steps, so you multiply by r three times, and r^3 = 324 ÷ 12 = 27. The cube root of 27 is 3, so r = 3. Then work back to the first term: a = 12 ÷ 3 = 4. Check by building the sequence 4, 12, 36, 108, 324. The 2nd term is 12 and the 5th is 324, as required.

A different style puts an unknown letter in the middle of three terms, such as 6, x, 54. In a geometric sequence the ratio from the first to the second equals the ratio from the second to the third, so x/6 = 54/x. Cross-multiplying gives x^2 = 6 × 54 = 324, so x = 18 or x = -18. Read the question: if it says the terms are positive, only x = 18 is allowed, and the ratio is 3. If it gives no such information, both values make a geometric sequence, and you should give both.

Method: 1. Count the number of steps k between the two known terms. 2. Divide the later term by the earlier term to get r^k, then take the kth root. 3. Work back to a by dividing the known term by r the right number of times, then check.

Growth and decay in context

Percentage change is a geometric sequence. If a sum of money grows by 4% each year, each year’s total is 1.04 times the year before, so the ratio is r = 1.04. If a machine loses 20% of its value each year, the ratio is 0.8. The multiplier is 1 plus the rate for growth and 1 minus the rate for decay. The value now is the first term, and the value after n years is the term in position n + 1, because year 0 is the first term.

This is the reason the exponent seems to be off by one. After 5 years the term is u_6 = a r^5. For a deposit of pounds 2000 at 4% a year, the values are pounds 2000, pounds 2080, pounds 2163.20, and so on. After 5 years the value is 2000 × 1.04^5 = pounds 2433.31 to the nearest penny. Write the multiplier explicitly in your working, and round only at the end. Questions on compound interest and depreciation use exactly this idea.

Method: 1. Turn the percentage change into a multiplier: 1.04 for 4% growth, 0.8 for 20% decay. 2. The starting amount is the first term. After n steps use the power n on the multiplier. 3. Round to the accuracy asked for only after the final calculation.