Find the sum of `2n` terms of the series whose every even term is `' a '` times the term before it and every od term is `' c '` times the term before it, the first term being unity.

Options

A. `((1-a)(1-c^(n)a^(n)))/(1-ca)`
B. `((1-a)(1-c^(n-1)a^(n-1)))/(1-ca)`
C. `((1-a)(1-c^(n-2)a^(n-2)))/(1-ca)`
D. none of these

Answer

D

Explanation

According to the given condition the given series is
1+a+ca+a(ca)+c(aca)+ . . .to 2n terms.
`=(1+ca+c^(2)a^(2)+ . . .` to n terms) `+(a+ca^(2)+c^(2)a^(3)+ . . . ` to n terms)
`={(1-(ca)^(n))/(1-ca)}+a{(1-(ca)^(n))/(1-ca)}=({1-(ca)^(n)}(1+a))/(1-ca)`