Question

Find the sum to n terms of the A.P. whose $ {{k}^{th}} $ term is $ 5k+1 $ .

Answer 1

Hint: From the given series of arithmetic sequences, we find the general term of the series. We find the formula for $ {{t}_{k}} $ , the $ {{k}^{th}} $ term of the series. From the given sequence we find the common difference between the two consecutive terms. We put the values to get the formula for the general term $ {{t}_{n}} $ and the formula of summation. Then we put the value in the formula to find the sum to n terms.

Complete step by step solution:
We express the arithmetic sequence in its general form.
We express the terms as $ {{t}_{k}}={{k}^{th}} $ term which is $ 5k+1 $ .
Putting the values of $ k=1,2,3,..... $ we get the series.
The series is $ 6,11,16,.... $ .
The first term be $ {{t}_{1}} $ and the common difference be $ d $ where $ d={{t}_{2}}-{{t}_{1}}={{t}_{3}}-{{t}_{2}}={{t}_{4}}-{{t}_{3}} $ .
The first term is 6. So, $ {{t}_{1}}=6 $ . The common difference is $ d={{t}_{2}}-{{t}_{1}}=11-6=5 $ .
Now we need to find the formula of summation of the arithmetic sequence.
The general formula for n terms is \[{{S}_{n}}=\dfrac{n}{2}\left[ 2{{t}_{1}}+\left( n-1 \right)d \right]\].
There are n terms in the series $ 6,11,16,.... $
The sum of the series will be \[{{S}_{n}}=\dfrac{n}{2}\left[ 2\times 6+5\left( n-1 \right) \right]=\dfrac{n\left( 5n+7 \right)}{2}\]
Therefore, the sum of each arithmetic series whose $ {{k}^{th}} $ term is $ 5k+1 $ , is \[\dfrac{n\left( 5n+7 \right)}{2}\].
So, the correct answer is “\[\dfrac{n\left( 5n+7 \right)}{2}\]”.

Note: The sequence is an increasing sequence where the common difference is a positive number. The common difference will never be calculated according to the difference of greater number from the lesser number.