Question

In an A.P of 21 terms, the sum of the first three term is 33 and that of middle 3 is 75. What is sum of A.P

Answer 1

Say, the middle of of the 21 terms be A

& the term difference be r.

So, the A. P is,

A-10r , A-9r , A-8r , A-7r , A-6r , A-5r , A-4r , A-3r , A-2r , A-r , A , A+r , A+2r , A+3r , A+4r , A+5r , A+6r , A+7r , A+8r , A+9r , A+10r

As per your question,

$( a-10 r) + (a -9 r) + ( a-8 r) = 33$

&

$( a- r) + a + (a +r ) = 75$

from the second equation we get,

$3a = 75$

or,

$a = 25$

If you notice that , the summation formula of the A. P is 21A (Irrespective of the value of r)

so, the answer is

$21 \times 25 = 525$