Question

There are 37 terms in an A.P. the sum of the middle three terms is 225 and the sum of the last three terms is 429. Find A.P.

Answer 1

First find the middle term and then form two equations using the two given conditions. This will help in finding the value of common difference d. then using this value of d find the A.P.

Complete step by step answer:

Given that there are 37 terms.
So n=37.
Middle term = $\dfrac{{n + 1}}{2} = \dfrac{{37 + 1}}{2} = \dfrac{{38}}{2} = 19$
In order to find ${n^{th}}$ term in an A.P. we use formula $a + \left( {n - 1} \right)d$.
Three middle terms are${18^{th}},{19^{th}},{20^{th}}$.
But in an A.P. we can write,
${18^{th}} = a + 17d$
${19^{th}} = a + 18d$
${20^{th}} = a + 19d$

Similarly, we can write for the last three terms

$${35^{th}} = a + 34d \\\ {36^{th}} = a + 35d \\\ {37^{th}} = a + 36d \\\ \\\$$

The Sum of the three middle terms is 225.

$$\Rightarrow a + 17d + a + 18d + a + 19d = 225 \\\ \Rightarrow 3a + 54d = 225 \\\$$

Sum of last three terms is 429.

$$\Rightarrow a + 34d + a + 35d + a + 36d = 429 \\\ \Rightarrow 3a + 105d = 429 \\\$$

Now from these two equations find the value of d.

$$\Rightarrow 3a + 105d - \left( {3a + 54d} \right) = 429 - 225 \\\ \Rightarrow 105d - 54d = 204 \\\ \Rightarrow 51d = 204 \\\ \Rightarrow d = \dfrac{{204}}{{51}} \\\ \Rightarrow d = 4 \\\$$

Now putting this value in any of the above equation we can find value of a.

$$\Rightarrow 3a + 54d = 225 \\\ \Rightarrow 3a + 54 \times 4 = 225 \\\ \Rightarrow 3a = 225 - 216 \\\ \Rightarrow 3a = 9 \\\ \Rightarrow a = 3 \\\$$

Since we have the value of the first term and that of common difference also. So let’s find the A.P.
Therefore the A.P. series is $3,7,11,15,....$

Note: Finding the middle term is the important step of the problem because the whole problem is dependent on it. Once we get the middle term we can use the conditions given in the problem.
Additional information: An arithmetic progression is a series in which two terms are having a common difference d between them. a, a+d, a+2d, ….is an A.P. Sum of first n terms in an A.P. is given by the formula ${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$.