Question

Sum of ${4^{{\text{th }}}}$ and ${8^{{\text{th }}}}$ terms of an AP is 24 and the Sum of ${6^{{\text{th }}}}$and ${10^{{\text{th }}}}$terms of AP is 44. Find the first three terms of AP, also find the sum of 50 terms of AP.

Answer 1

Hint:
We solve this using the general term of AP which is ${a_n} = a + (n - 1) \cdot d$ and find the sum using general formula of sum of AP given as: ${S_n} =\dfrac{n}{2}[2a + (n - 1)d]$

Complete step by step solution:
Given that Sum of ${4^{{\text{th }}}}$ and ${8^{{\text{th}}}}$ term is 24.
Now ${4^{{\text{th}}}}$ term is given in general equation by putting n=4, we get,
${a_n} = a + (n - 1)d \Rightarrow {a_4} = a + (4 - 1)d = a + 3d$
Similarly,${8^{{\text{th}}}}$ term is given in general equation by putting n=8, we get,
${a_8} = a + (8 - 1)d \Rightarrow {a_8} = a + 7d$
Now Sum of ${4^{{\text{th }}}}$and ${8^{{\text{th}}}}$ term is given as,
${a_4} + {a_8} = 24 \Rightarrow a + 3d + a + 7d = 24 \Rightarrow 2a + 10d = 24$
$ \Rightarrow a + 5d = 12$--$(A)$
Similarly, given that Sum of ${6^{{\text{th }}}}$ and ${10^{{\text{th}}}}$ term is 44.
${a_6} = a + (6 - 1)d$
$ \Rightarrow {a_6} = a + 5d$
${a_{10}} = a + (10 - 1)d = a + 9d$
Now, ${a_6} + {a_{10}} = 44 \Rightarrow a + 5d + a + 9d = 44 \Rightarrow 2a + 14d = 44$
$ \Rightarrow a + 7d = 22$ --$(B)$
Solve (A) and (B): (A) - (B)
$a + 5d - a - 7d = 12 - 22$
$ \Rightarrow - 2d = - 10$
$ \Rightarrow d = 5.$, Now put value of d in (A)
$ \Rightarrow a = 2$
Now, first three terms of AP given as $ = a,a + d,a + 2d$
$ \Rightarrow 2,2 + 5,2 + 2 \times 5 = 2,7,10$
First three terms of AP = $2,7,10$
Now, let’s find the sum of first 50 terms of AP
The general formula of sum of series in AP given as:
${S_n} =\dfrac{n}{2}[2a + (n - 1)d]$
Put n = 50, for the sum of first 50 terms of AP
$ \Rightarrow {S_{50}} =\dfrac{{50}}{2}[2 \times 2 + (50 - 1) \times 5]$
$ \Rightarrow {S_{50}} = 25[4 + 49 \times 5] = 25 \times [4 + 245] = 25 \times 249$
$ = 6225$
Therefore, Sum of the first 50 terms of AP = 6225.

First three terms of AP = $2,7,10$
Sum of first 50 terms of AP = 6225

Note:
While solving don’t go for the sum first because when you find the value of a (first term) and d(common difference) you can find the sum of terms for that you have to first find out the a and d in this question situation. So, just forget first the 2nd question is also given that is find the sum. Just do questions step by step.