Number Systems -Sequence and Series

[sukhman]

In the infinite sequence A, An = x ^(n - 1) + x^n + x^ (n + 1 ) + x ^ (n + 2)+ x^ (n + 3), where x is a positive integer constant. For what value of n is the ratio of An to x(1 + x(1 + x(1 + x(1 + x)))) equal to x^5 ? (A) 8 (B) 7 (C) 6 (D) 5 (E) 4

[ganeshrkamath]

In the infinite sequence A, An = x ^(n - 1) + x^n + x^ (n + 1 ) + x ^ (n + 2)+ x^ (n + 3), where x is a positive integer constant. For what value of n is the ratio of An to x(1 + x(1 + x(1 + x(1 + x)))) equal to x^5 ? (A) 8 (B) 7 (C) 6 (D) 5 (E) 4

The highest power of x in x(1 + x(1 + x(1 + x(1 + x)))) is 5.
The highest power of x in A(n) is (n+3)
When A(n) is divided by the given term, we get x^5.

(n+3) = 5 + 5
n = 7

Choose B

Cheers

[sukhman]

Sorry put the wrong statement in wrong post!!

[ganeshrkamath]

I did not understand what I got was 3^11.5^11=5*x^11 is this equation right ?

Can you elaborate your method?

Alternate approach:
An = (x(1 + x(1 + x(1 + x(1 + x))))) * x^5
= (x(1 + x(1 + x(1 + x + x^2)))) * x^5
= (x(1 + x(1 + x + x^2 + x^3))) * x^5
= (x(1 + x + x^2 + x^3 + x^4)) * x^5
= (x + x^2 + x^3 + x^4 + x^5)) * x^5
= x^6 + x^7 + x^8 + x^9 + x^10

Also,
An = x ^(n - 1) + x^n + x^ (n + 1 ) + x ^ (n + 2)+ x^ (n + 3)

So (n+3) = 10
n = 7

Hope this helps.

Cheers