Question

Find the value of ${20^{th}}$ term from the last term of the AP: 3, 8, 13………, 253.

Answer 1

Hint: In order to solve this question we have to find the total number of terms by using the formula ${a_n} = a + \left( {n - 1} \right)d$ then we get the ${20^{th}}$ term from last by using the formula-
${m^{th}}$ from the last= total terms $ - {m^{th}}$ +1.

Complete step-by-step answer:
Given series are
3, 8, 13………, 253
And we have to find the ${20^{th}}$ term from the last term of this series.
Here a=3 and d=8-3=5
${a_n}$ = 253
Let n be the total number of terms at first.
Now we know that
${a_n} = a + \left( {n - 1} \right)d$-------(1)
Putting the values of a, d, ${a_n}$ we get,
253=3+(n-1)$ \times $ 5
253-3=(n-1)$ \times $ 5
$\dfrac{{250}}{5}$ =(n-1)
50=n-1
n=50+1
n=51
Thus there are 51 terms in the given series.
Now we know that,
${20^{th}}$ from the last= total terms- ${20^{th}}$ +1
So, 20th term from last = 51-20+1=32
Hence, 20th term from the last = ${32^{th}}$ term from first
Or we know that,
${a_n} = a + \left( {n - 1} \right)d$
${a_{32}}$ = 3+(32-1)$ \times $ 5
Or ${a_{32}}$ = 158
Thus, ${20^{th}}$ term from the last = 158

Note: Whenever such types of questions come in front of us the general mistake a person can do is putting wrong value in the formula. Moreover, this question can also be solved by another method i.e. by considering the last term first term and by using the formula ${a_n} = a + \left( {n - 1} \right)d$.