Question

26. An A.P. consists of n terms. If the sum of its first three terms
is x and the sum of the last three terms is y, then show that,
the sum of all the terms of the A.P. is : n/6(x + y).​

Answer 1

If we consider an odd no. of terms in an AP, then their sum will be equal to the product of the middle term and the no. of terms.

From the equation,

S_n = (a_1 + a_n)n / 2,

n is the no. of terms, we know, and (a_1 + a_n)/2 should be the middle term. This is possible only when n is odd.

Well, we've already studied in earlier classes that Sum = Average × No. of Terms!!!

So the sum of first three terms is 3 times the second term, i.e.,

x = 3 · a_2

x = 3(a + d)

=> a_1 = a = (x / 3) - d

And, since the last three terms are (n - 2)th, (n - 1)th and (n)th terms, their sum is,

y = 3 · a_(n - 1)

y = 3[a + (n - 2)d]

=> a_n = a + (n - 1)d = (y / 3) + d

Now, the sum of all terms,

S_n = (a_1 + a_n)n / 2

S_n = ((x / 3) - d + (y / 3) + d)n / 2

S_n = ((x + y) / 3) (n / 2)

S_n = (x + y)(n / 6)

Hence Proved!

Answer 2

Answer:

I hope it will help you...