Question

An A.P. consists of 37 terms. The sum of three middle most terms is 225 and the sum of last three terms is 429 then find the A.P.

Answer 1

According to given in the question we have to determine the A.P. when An A.P. consists of 37 terms. The sum of three middle terms is 225. So, first of all we have to let the first term and the common difference of the A.P and the common difference is as explained below:
Common difference: Common difference is the difference between the second and the first term and same as third term and second term and if the differences are the same then we can say that the terms are in A.P.
Now, we have to find the middle term with the help of the formula as mentioned below:

Formula used: $\Rightarrow \dfrac{{n + 1}}{2}..................(A)$
Now, as mentioned in the question, the sum of three middle terms is 225 so, we can determine the equation in the form of first and the common difference.
Now, as mentioned in the question that the sum of the last three terms is 429 so, we can obtain another expression in the form of first term and common difference.
Now, we have to solve both of the expressions obtained to determine the value of the first term and the common difference then we can easily determine the required A.P.

Complete step-by-step solution:
Step 1: First of all we have to let the first term and the common difference for the A.P as mentioned in the solution hint. Hence, let
$\Rightarrow$First term$= a$
$\Rightarrow$Common difference$= d$
Step 2: Now, we have to find the middle term of the A.P with the help of the formula (A) as mentioned in the solution hint. Hence,
Middle term
$= \dfrac{{37 + 1}}{2} \\\ = \dfrac{{38}}{2} \\\ = {19^{th}}term$
Step 3: Now, we can determine all the three terms as with the help of the solution step 2. Hence, the terms are ${18^{th}},{19^{th}}$ and ${20^{th}}$
${18^{th}}$term = $(a + 17d)$
${19^{th}}$ term = $(a + 18d)$ and,
${20^{th}}$ term =$(a + 19d)$
Step 4: Now, as mentioned in the question that the sum of all the three middle terms are 225 so, we can determine the sum and can obtain the expression in the form of first term and common difference. Hence,

$$\Rightarrow (a + 17d) + (a + 18d) + (a + 19d) = 225 \\\ \Rightarrow 3(a + 18d) = 225 \\\ \Rightarrow a + 18d = \dfrac{{225}}{3} \\\ \Rightarrow a + 18d = 75..............(1)$$

Step 5: Now, same as the step 4 we have to determine the sum of all the last three terms which is 429. Hence, all the last three terms are ${35^{th}},{36^{th}}$and ${37^{th}}$so, there sum are:
$\Rightarrow (a + 34d) + (a + 35d) + (a + 36d) = 429 \\\ \Rightarrow 3(a + 35d) = 429 \\\ \Rightarrow a + 35d = \dfrac{{429}}{3} \\\ \Rightarrow a + 35d = 143..............(2)$
Step 6: Now, to obtain the value of the first term and common difference we have to solve the both of the expressions by subtracting both of them. Hence,
$\Rightarrow (a + 18d) - (a + 35d) = 75 - 143 \\\ \Rightarrow 18d - 35d = 68 \\\ \Rightarrow d = \dfrac{{68}}{{17}} \\\ \Rightarrow d = 4$
Step 7: Now, to obtain the value of the first term we have to substitute the value of d in the expression (1). Hence,
$\Rightarrow a + 18(4) = 75 \\\ \Rightarrow a = 75 - 72 \\\ \Rightarrow a = 3$
Step 8: Now, with the help of first term and the common difference we can determine the required A.P. Hence,
A.P.$= 3,7,11,15,...........$

Hence, with the help of the formula (A) we have to determine the required A.P which is A.P.$= 3,7,11,15,...........$

Note: We can determine the common difference for the given A.P series by finding the difference between the second and first term and by finding the difference between the third and the second term which are equal to each other.
To determine the sum of all the three middle terms first of all we have to determine all the three terms but before that we have to determine the middle term which can be obtained with the help of the formula (A) as mentioned in the solution hint.