Question

What are the next three terms of the sequence: `-8, 24, -72, 216`?

Answer 1

`-648 , 1944 , -5832`

Explanation:

If the numbers:

`a , b , c` are a geometric sequence, then:

`b/a=c/b`

From our example:

`24/(-8)=(-72)/24=-3`

The result above is known as the common ratio.

The nth term of a geometric sequence is given by:

`ar^(n-1)`

Where `a` is the first term, `r` is the common ratio and `n` is the nth term.

So we have:

`-8(-3)^(n-1)`

We are seeking the 5th, 6th and 7th terms, so:

`n=5` , `n=6` and `n=7`

5th term `=-8(-3)^(5-1)=-8(-3)^4=-8(81)=-648`

6th term `=-8(-3)^(6-1)=-8(-3)^5=-8(-243)=1944`

7th term `=-8(-3)^(7-1)=-8(-3)^6=-8(729)=-5832`

Answer 2

`a_5 = -648, a_6 = 1944, a_7 = -5832`

Explanation:

Common ratio (r) of G S is

`r = a_2 / a_1 = a_3 / a_2 = a_4 / a_3 ....` where `a_1, a_2, a_3, a_4,...` are the terms 1, 2, 3, 4 ... respectively.

Given : `a_1 = -8, a_2 = 24, a_3 = -72, a_4 = 216, ...`

`:. r = 24 / -8 = -72 / 24 = 216 / -72, ...`

`r = -3`

`n^(th)` term is given by the formula `a_1 r^(n-1)`

`5^(th) term a_5 = a_1 * r^(5-1) = a r^4 = -8 * (-3)^4 = -648`

`6^(th) term a_6 = a_1 * r^(6-1) = a r^5 = -8 * (-3)^5 = 1944`

`7^(th) term a_7 = a_1 r^(7-1)`

`a_7 = a_1 * r^(6-1 + 1) = a_1 r^(6-1) * r color(red)(= a_6 * r) = 1944 * -3 = -5832`