Question

Without adding find the sum 1 + 3 + 5 + 7 +..........23.

Answer 1

all odd no.s from 1 to 23

are 12 ..

so without adding we will take the square 12 that is 144

You can check by adding.

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Answer 2

Tn = a + (n - 1) d,

where Tn = nth term and a = first term. Here d = common difference = Tn - Tn-1.

23 = 1 + (n - 1 ) 2

n = 12

Sum of first n terms of an AP: S =(n/2)[2a + (n- 1)d]

sum = 6{2 + 11×2} = 144

hope it helps