Question

Find the sum of a finite geometric sequence from n = 1 to n = 7, using the expression −4(6)n − 1.

111,325
526
782
-223,948

Answer 1

-223,948

Explanation:

The formula of a sum of terms of a gometric sequence:

`S_n=a_1\cdot(1-r^n)/(1-r)`

a₁ - first term

r - common ratio

We have

`a_n=-4(6)^(n-1)`

Calculate a₁. Put n = 1:

`a_1=-4(6)^(1-1)=-4(6)^0=-4(1)=-4`

Calculate the common ratio:

`r=(a_(n+1))/(a_n)\\\\a_(n+1)=-4(6)^(n+1-1)=-4(6)^n\\\\r=(-4(6)^n)/(-4(6)^(n-1))=6^n:6^(n-1)\\\\\text{use}\ a^n:a^m=a^(n-m)\\\\r=6^(n-(n-1))=6^(n-n+1)=6^1=6`

`\text{Substitute}\ a_1=-4,\ n=7,\ r=6:\\\\S_7=-4\cdot(1-6^7)/(1-6)=-4\cdot(1-279936)/(-5)=-4\cdot(-279935)/(-5)=(-4)(55987)\\\\S_7=-223948`