Question

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

Answer 1

Answer:

-223,948

Step-by-step explanation:

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

$S_n=a_1\cdot\dfrac{1-r^n}{1-r}$

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:

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

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