Question

If the nth term of an (2n+1)find the sum of first n term of AP

Answer 1

a+(n-1)d = 2n+1
sum= (n/2)*(a+2n+1)
= an/2+n^2 + +n/2
hope it helped u

Answer 2

Heya user,

Clearly, if nth term = 2n + 1;
We have $T_1$ = 2(1) + 1 = 3
Also, $T_2$ = 2(2) + 1 = 5
=> $T_2$ - $T_1$ = 5 - 3 = 2
Hence, the common difference = 2 and the first term = 3......

Now,
Sum of first n terms = [ 3 + 5 + 7 + ... + n ]
-> "   "    "     "    "      = [ 1 + 3 + 5 + ... + n ] - 1

Also, we know sum of 'n' odd terms is n²  <---- (1)
=>Sum of first n terms = n² - 1; 

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Further, for your convenience, I've provided the proof of our (1) claim:

a = 1;
d = 2;

=> $S_n$ = n/2 [ 2(a) + (n-1)d ] = n/2 [ 2 + (n-1)2 ] = n/2 [ 2 + 2n - 2 ]
=> $S_n$ = n/2 [ 2n ] = n²

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