Question

Find the number of terms of the A.P $54,51,48,.......$ so that their sum is $513$ . Explain the double answer.

Answer 1

Hint: Here we have been given an A.P and we have to find the terms of it such that the sum of them is equal to $513$ . Firstly we will write the common difference and the first value of the A.P then we will write the sum of an A.P formula. Finally we will put the value in the formula and get the desired answer.

Complete step-by-step solution:
The A.P is given as follows:
$54,51,48,.....$
The sum is given as,
$513$…..$\left( 1 \right)$
The first of the A.P is
First Term $a=54$….$\left( 2 \right)$
As we can see that it is a decreasing A.P so we will subtract the second term by the first term to get the common difference as follows:
Common difference $d=51-54$
$\Rightarrow d=-3$ …..$\left( 3 \right)$
Now as we know the sum of an A.P is calculated as follows:
${{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$
On substituting the value from equation (1), (2) and (3) above we get,
$\Rightarrow 513=\dfrac{n}{2}\left[ 2\times 54+\left( n-1 \right)\times -3 \right]$
$\Rightarrow 513\times 2=n\left[ 108-3n+3 \right]$
Simplifying further we get,
$\Rightarrow 1026=108n-3{{n}^{2}}+3n$
$\Rightarrow 1026=111n-3{{n}^{2}}$….$\left( 4 \right)$
Take all values on one side,
$\Rightarrow 3{{n}^{2}}-111n+1026=0$
Divide whole value by $3$ ,
$\Rightarrow {{n}^{2}}-37n+342=0$
Using splitting the middle term way we get,
$\Rightarrow {{n}^{2}}-18n-19n+342$
$\Rightarrow n\left( n-18 \right)-19\left( n-18 \right)=0$
Take common outside from the two values,
$\Rightarrow \left( n-18 \right)\left( n-19 \right)=0$
So we got the two values as
$n=18,19$
Hence for both $18$ term and $19$ term we will get the sum as $513$ .
Next we will show why we are getting two answers.
We will use the formula to find the ${{n}^{th}}$ term,
${{T}_{18}}=a+17d$
$\Rightarrow {{T}_{18}}=54+17\times -3$
So our ${{18}^{th}}$ term
${{T}_{18}}=3$
Next our ${{19}^{th}}$ term will be,
${{T}_{19}}=a+18d$
$\Rightarrow {{T}_{19}}=54+18\times -3$
So our ${{19}^{th}}$ term is,
${{T}_{19}}=0$
As we are getting the ${{19}^{th}}$ term as $0$ that is the reason we are getting double answers.

Note: Arithmetic progression (A.P) is a sequence of numbers such that the difference between the consecutive numbers is always the same. If we know the common difference and the first term it becomes very easy to find the sum of the A.P and also to find any number of an A.P. The finite portion of the Arithmetic progression is known as finite arithmetic progression.