Question

What is the first term in a geometric sequence if the common ratio is −3 and the sum of the first six terms is 1,274?

Answer 1

Sum of the first n terms in a geometric sequence:

                                   1 - r^n
Sum of n terms = A1 -------------
                                      1 - r

Here r = - 3 , n = 6 and sum of first 6 terms = 1274

                          1 - (-3)^6              1 - 729
=> 1274 = A1   -------------- =  A1   -----------
                            1 - (-3)                  1 + 3

1274 = A1 (- 728 / 4)

=> A1 = - 1274 * 4 / 728 = - 7

Verification:

n       An

1       - 7
2       - 7 * (-3) = 21
3       21 * (-3) = - 63
4       - 63 * (-3) = 189
5       189 * (-3) = - 567
6      - 567 * (-3) = 1701

Sum = -7 + 21 - 63 + 189 - 567 + 1701 = 1274

Answer: - 7