Question

Evaluate (precalculus)

Answer 1

S∞ = 7 Option C

Explanation:

Given that:

∑21
`((1)/(4) )^(i)` i = 1 to ∞

We need to find the sum.

Firstly we will find the sequence

a₁ =
`21((1)/(4) )^(1)  = ((21)/(4) )`

a₂ =
`21((1)/(4) )^(2)  = ((21)/(16) )`

a₃ =
`21((1)/(4) )^(3)  = ((21)/(64) )`

a₄ =
`21((1)/(4) )^(4)  = ((21)/(256) )`

this sequence is geometric

So, we use sum of infinite terms

a =
`(21)/(4)`

r =
`((21)/(16) )/((21)/(4) )  = (1)/(4)`

The sum of infinite terms of a GP series

S∞ =
`(a)/(1-r)`

where a is first term

r is common ratio and 0<r<1

S∞ =
`((21)/(4) )/(1 - (1)/(4) )  = ((21)/(4) )/((3)/(4) )`

S∞ = 7