## Comparing consecutive terms of a series

The question is based upon sequence and finding the relation between consecutive terms of a series with its next or previous term if the expression of its nth term is given.

## Try the problem

Let \(a_n = \frac{{10^{n+1}+1}}{10^n +1}\), for \(n=1,2,3……\) . Then

(A) for every \(n, a_n \geq a_{n+1} ;\)

(B) for every \(n, a_n \leq a_{n+1} ;\)

(C) there is an integer k such that \(a_{n+k}=a_n\) for all n.

(D) None of the above.

I.S.I. Entrance B. stat. 2003, Objective, Problem 1

Sequence and series

6 out of 10

Secrets in mathematics.

## Knowledge Graph

## Use some hints

First hint

we can put value of n , as it is equal to list of given natural numbers. n=1,2,3,4,……… and verify the result with the option.

Second Hint

We can see weather the series is converging or diverging to a value, means is there any pattern of the next term with the previous term in the sense of ratio of the two terms.

Final Step

Also to generalize everything we can put n+1 in place of n in the expression \(a_n = \frac{{10^{n+1}+1}}{10^n +1}\) and then we can divide the \(a_{n+1}\) by \(a_{n}\) to get the required comparison ratio.

## Other Useful links

- https://www.cheenta.com/slope-of-straight-line-amc-10b-2012-problem-3/
- https://www.youtube.com/watch?v=9gPEKehjxr8