Question

The geometric sequence a_ia i ​ a, start subscript, i, end subscript is defined by the formula: a_1 = -\dfrac13a 1 ​ =− 3 1 ​ a, start subscript, 1, end subscript, equals, minus, start fraction, 1, divided by, 3, end fraction a_i = a_{i - 1} \cdot (-3)a i ​ =a i−1 ​ ⋅(−3)a, start subscript, i, end subscript, equals, a, start subscript, i, minus, 1, end subscript, dot, left parenthesis, minus, 3, right parenthesis Find the sum of the first 757575 terms in the sequence.

Answer 1

Answer:

1.014*10^35

Step-by-step explanation:

Given the following geometric sequence

$a_1 = \frac{-1}{3} \\a_i = a_{i-1}*(-3)$

Sum of nth term of the GP is expressed as;

$S_n = \frac{a(1-r^n)}{1-r}$

a = -1/3

n = 75

r = a2/a1

a2 = -3a1

a2 = -3(-1/3)

a2 = 1

r = 1/(-1/3)

r = -3

Substitute the given values in the formula;

$S_n = \frac{-1/3(3^{75} - 1)}{3-1}\\S_n = \frac{3^{75}-1}{-6} \\S_{75} = 1.014*10^{35}$

Hence the sum of the first 75 terms is 1.014*10^35