Question

Find the sum of first 24 term of the ap a1,a2,a3....... if it is known that a1+a5+a10+a15+a20+a24=225

Answer 1

AnswEr:

We know that in an A.P. the sum of the terms equidistant from the beginning and end is always same and is equal to the sum of first and last term

$\sf \: i.e. \: a_1 + a_n = a_2 + a_n - 1 = a_3 + a_n \\ \sf - 2 = ...$

So, if an A.P. consists of 24 term , then

$\sf \: a_1 + a_24 = a_5 + a_20 = a_10 + a_15.$

_________

Now,

$\sf \: a_1 + a_5 + a_10 + a_15 + a_20 + a_24 = 225 \\ \\ \implies \sf(a_1 + a_24) + (a_5 + a_20) + (a_10 \\ \sf + a_15) = 225 \\ \\ \implies \sf \: 3(a_1 + a_24) = 225 \\ \\ \implies \sf \: a_1 + a_24 = \frac{225}{3} = 75 \\ \\ \therefore \sf \: S_24 = \frac{24}{2} (a_1 + a_24) \\ \\ \implies \sf \: S_24 = 12(75) = 900$