Question

If the sum of first 20 terms of AP is equal to sum of first 30 terms of AP then the sum of first 50 terms is

Answer 1

Here we have to find the sum of the first 50 terms of an AP, for that, we will first find the sum of 20 terms using the formula of sum of an AP and we will first find the sum of 30 terms of an AP using the formula of sum of an AP. We will then equate the sum of 20 terms of an AP with the sum of 30 terms of an AP. From there, we will get the relation between the first term and the common difference of an AP. Then we will first find the value of sum of 20 terms using the formula of sum of an AP and also we will use the relation obtained between the first term and the common difference of an AP here.

Complete step by step solution:
We know the formula of sum of an AP is given by
${{S}_{n}}=\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)$
Now, we will find the sum of the first 20 terms of an AP using this formula of sum of an AP.
$\Rightarrow {{S}_{20}}=\dfrac{20}{2}\left( 2a+\left( 20-1 \right)d \right)$
On simplifying the terms, we get
$\Rightarrow {{S}_{20}}=20a+190d$ ……… $\left( 1 \right)$
Now, we will find the sum of the first 30 terms of an AP using this formula of sum of an AP.
$\Rightarrow {{S}_{30}}=\dfrac{30}{2}\left( 2a+\left( 30-1 \right)d \right)$
On simplifying the terms, we get
$\Rightarrow {{S}_{30}}=30a+435d$ ………
Equating equation 1 and equation 2, we get
$\Rightarrow 30a+435d=20a+190d$
On adding and subtracting the like terms, we get
$\Rightarrow 10a+245d=0$
Dividing both sides by 5, we get
$\Rightarrow 2a+49d=0$ ………. $\left( 3 \right)$
Now, we will find the sum of the first 50 terms of an AP using the formula of sum of an AP.
$\Rightarrow {{S}_{50}}=\dfrac{50}{2}\left( 2a+\left( 50-1 \right)d \right)$
On simplifying the terms, we get
$\Rightarrow {{S}_{50}}=50a+1225d$
Taking 25 common from right side of equation, we get
$\Rightarrow {{S}_{50}}=25\left( 2a+49d \right)$
We know the value $2a+49d=0$ from equation 3.
On substituting this value here, we get
$\Rightarrow {{S}_{50}}=25\times 0=0$

Thus, the sum of the first 50 terms of this AP is equal to zero.

Note:
We have obtained the sum of an arithmetic progression. An arithmetic progression is defined as a sequence in which the difference between the term and the preceding term is constant or in other words, we can say that an arithmetic progression is a sequence such that every element after the first is obtained by adding a constant term to the preceding element.