Question

a 1 ​ =− 3 1 ​ a, start subscript, 1, end subscript, equals, minus, start fraction, 1, divided by, 3, end fraction a i = a i − 1 ⋅ ( − 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 75 7575 terms in the sequence.

Answer 1

`S_(75)=-506.888989761$ X 10^(32)`

Explanation:

Given the sequence

`a_1=-(1)/(3)\\ a_i=a_(i-1)\cdot (-3)`

`a_2=a_(2-1)\cdot (-3)=a_1 \cdot (-3) = -(1)/(3)\cdot (-3)=1\\a_3=a_(3-1)\cdot (-3)=a_2 \cdot (-3) = 1 \cdot (-3)=-3\\a_4=a_(4-1)\cdot (-3)=a_3 \cdot (-3) = -3 \cdot (-3)=9\\`

Therefore the sequence is:

`-(1)/(3), 1, -3, 9, ...`

This is a geometric sequence where the:

First Term,
`a_1=-(1)/(3)`

Common ratio, r =-3

We want to determine the sum of the first 75 terms.

For a geometric sequence, the sum:

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

Therefore:

`S_(75)=(-(1)/(3)(1-(-3)^(75)))/(1-(-3)) \\\\=(-(1-(-3)^(75)))/(4*3)\\\\S_(75)=((-3)^(75)-1)/(12)\\\\S_(75)=-506.888989761$ X 10^(32)`