Question

Given the first and the last terms of a seven-term geometric series with a negative common ratio, what is the sum of all the terms?
2-...+128
a) 258
b) 129
c) 86
d) 43

Answer 1

By using the general formula for a geometric sequence, we will find that the correct option is c: 86

Working with geometric sequences:

The general formula for the nth term in a geometric sequence is:

$a_n = a_1*r^{n-1}$

Where r is the common ratio of the geometric sequence here we know that:

• a₁ = 2
• a₇ = 128
• r is negative.

Replacing the first two in the general formula we get:

$a_7 = a_1*r^6\\ \\ 128 = 2*r^6\\\\ 128/2 = 64 = r^6\\ \\ -64^{1/6} = r = -2$

Where we took the negative solution for the root. Now that we know the common ratio we can find the other terms:

$a_1 = 2\\ a_2 = 2*(-2) = -4\\ a_3 = -4*(-2) = 8\\ a_4 = 8*(-2) = -16\\ a_5 = -16*(-2) = 32\\ a_6 = 32*(-2) = -64\\ a_7 = -64*(-2) = 128$

The sum gives:

2 - 4 + 8 - 16 + 32 - 64 + 128 = 86

So the correct option is c.

If you want to learn more about geometric sequences, you can read:

brainly.com/question/9300199