Question

An A.P. 5, 12, 19 has 50 terms. Find its last term and hence find the sum of the terms.

Answer 1

Hint: From the given series of arithmetic sequence, we find the general term of the series. We find the formula for $ {{t}_{n}} $ , the $ {{n}^{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 formula of summation. Then we put the value in the formula to find the solution.

Complete step-by-step answer:
We have been given a series of arithmetic sequence which is $ 5,12,19,..... $
We express the arithmetic sequence in its general form.
We express the terms as $ {{t}_{n}} $ , the $ {{n}^{th}} $ term of the series.
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}} $ .
We can express the general term $ {{t}_{n}} $ based on the first term and the common difference.
The formula being $ {{t}_{n}}={{t}_{1}}+\left( n-1 \right)d $ .
The first term is 5. So, $ {{t}_{1}}=5 $ . The common difference is $ d={{t}_{2}}-{{t}_{1}}=12-5=7 $ .
We express general term
 $ {{t}_{n}} $ as $ {{t}_{n}}={{t}_{1}}+\left( n-1 \right)d=5+7\left( n-1 \right)=7n-2 $ .
We need to find the $ {{t}_{50}} $ , the $ {{50}^{th}} $ term which is $ {{t}_{50}}=7\times 50-2=348 $ .
Now we need to find the formula of summation of the arithmetic sequence.
The general formula for n terms is
\[{{S}_{n}}=\dfrac{n}{2}\left[ 2{{t}_{1}}+\left( n-1 \right)d \right]\].
We need to find the sum of the 50 terms of the series $ 5,12,19,..... $ .
The sum of the series will be \[{{S}_{50}}=\dfrac{50}{2}\left[ 2\times 5+7\left( 50-1 \right) \right]=25\times 353=8825\]
Therefore, the sum of each arithmetic series $ 5,12,19,..... $ is 8825.
So, the correct answer is “8825”.

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.