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## SAT Geometric Progression : Finding nth term of a GP

### Question 3: 6th term of a Geometric Sequence

What is the 6th term of a geometric sequence if the difference between its 3rd and 1st term is 9 and that between its 4th and 2nd term is 18?

1. 24
2. 3
3. 2
4. 48
5. 96
Choice E. The 6th term of the GP is 96.

This is an easy to moderate level difficulty SAT Math Question. The formula to find the nth term of a GP is given in the Hint / Formula tab. Use the formula and the steps outlined to solve the question. If you need further assistance, step wise explanation is provided in the subsequent tabs.

The nth term of a geometric sequence an = $a r^{n-1} $\$ Where a is the first term, 'n' is the number of terms and 'r' is the common ratio. To find the 6th term of the geometric sequence, you need the first term 'a' and the common ratio 'r'. Data given: Difference between the 3rd and the 1st term; the difference between the 4th and the 2nd term. 1. Express the difference between the 3rd and the 1st term in terms of the first term and the common ratio. 2. Express the difference between the 4th and the 2nd term in terms of the first term and the common ratio. 3. Divide the expression obtained in step 2 by the expression obtained in step 1 to determine common ratio. 4. Substitute the value of the common ratio in either step 1 or step 2 to determine the first term. 5. Use these two values to find the value of the 6th term. ### Step 1: Compute the common ratio 'r' from the data given Difference between the 3rd and the 1st term is 9. a3 - a = 9 The 3rd term of the sequence a3 = a * r3 - 1 = a * r2 Therefore, a3 - a = a * r2 - a = 9 or a$r2 - 1) = 9 .... eqn (1)

Difference between the 4th and the 2nd term is 18.

a4 - a4 = 9

The 4th term of the sequence a4 = a * r4 - 1 = a * r3

And the 2nd term of the sequence a2 = a * r2 - 1 = a * r

Therefore, a4 - a2 = a * r3 - ar = 18 or ar(r2 - 1) = 18 .... eqn (2)

Divide eqn(2) by eqn (1) to get r, the common ratio

$ar $lbrack r^2 - 1\rbrack \over {a \lbrack r^2 - 1\rbrack} \\$ = $18 \over 9\\$ or the common ratio r = 2 ### Step 2: Compute the first term 'a' and then the 6th term Substitute 'r' in eqn$1) to determine the first term 'a'

a(r2 - 1) = 9

a(22 - 1) = 9

or 3a = 9 or a = 3.

Last step: Compute the 6th term using a and r

a6 = a * r6 - 1 = a * r5

a6 = 3 * 25 = 3 * 32 = 96

## Concept Overview : Arithmetic Progression

Get a quick overview on the basic concepts of Arithmetic sequence including formulae to find the nth term of an arithmetic sequence and the sum of n terms of an arithmetic sequence.

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