Question

If the sum of the first 101 terms of an A.P. is 0 and if 1 is the first term of the A.P., then find the sum of the next 100 terms.
A.-101
B.201
C.-201
D.-200

Answer 1

Hint: First write the sum formula of A.P. Then put 101 for n, 1 for a and 0 as the sum to obtain the common difference. After that, obtain the sum of 201 terms then subtract the sum of 101 terms from it to obtain the sum of the next 100 terms.

Formula used:
Sum of n terms of an A.P is,
\[{S_n} = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right]\]

Complete step by step solution:
Substitute 101 for n and 1 for a in the equation \[{S_n} = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right]\]to obtain the value of d.
\[0 = \dfrac{{101}}{2}\left[ {2.1 + (101 - 1)d} \right]\]
\[100d + 2 = 0\]
\[d = - \dfrac{2}{{100}}\]
   \[ = - \dfrac{1}{{50}}\]
Now,
\[{S_{201}} = \dfrac{{201}}{2}\left[ {2.1 + (201 - 1)\left( { - \dfrac{1}{{50}}} \right)} \right]\]
\[ = \dfrac{{201}}{2}\left[ {2 - \dfrac{{200}}{{50}}} \right]\]
\[ = \dfrac{{201}}{2}\left[ {2 - 4} \right]\]
\[ = \dfrac{{201}}{2} \times \left( { - 2} \right)\]
=\[ - 201\]
Therefore, the sum of next 100 terms is \[{S_{201}} - {S_{101}}\], but \[{S_{101}}\] is 0.
Therefore, the sum of the next 100 terms is -201.

  The correct option is C.

Note Sometimes students state the answer as the sum of 201 terms but here we have asked to calculate the sum of the next 100 terms, here the sum of the first 101 terms is 0 so it will not bother the answer but if it is a non zero term then the answer will be incorrect.