CBSEi Class 10 Sequences (AP and GP) CORE

(4) Sum of first n terms of an AP
Let the first n terms of an AP be
=
a, a+d, a+2d, a+3d, …, [a+(n 2)d], [a+(n 1d]
Let
Note :
Sn = a + (a+d) + (a+2d) + ………………. + [a + (n 2)d] + [a + (n 1)d] (1)
Also, Sn = [a + (n 1)d] + [a + (n 2)d] + [a + (n 3)d] + …………. + (a+d) + a (2)
[Writing the terms in a reverse order]
Adding (1) and (2), term by term.
2Sn = [2a + (n-1)d] + [2a + (n-1)d] + ……… + [2a + (n-1)d]
n times
a + [a + (n 1) d] = 2a + (n 1) d
(a+d) + [a + (n 2) d] = 2a + (n 1) d
(a+2d) + [a + (n 3) d] = 2a + (n 1) d
[a + (n 2) d] + a+d] = 2a + (n 1)d
[a +(n 1) d] + a = 2a+ (n 1) d
2 Sn = n [2a + (n 1) d]
So, Sn = [2a + (n 1) d]
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