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Sequences & Series: Sum First Terms Common Difference First Term Integer

JEE Maths question with a full step-by-step solution.

Question
The sum of the first nn terms (n>1n>1) of an A.P. is 153153 and the common difference is 22. If the first term is an integer, the number of possible values of nn is
A33
B55correct
C44
D66
Solution
Step 1: With d=2d=2,
Sn=n2[2a+2(n−1)]=n(a+n−1)=153.S_n=\frac{n}{2}[2a+2(n-1)]=n(a+n-1)=153.
 Step 2: So n∣153=32⋅17n\mid 153=3^2\cdot 17 (then a=153n−n+1a=\frac{153}{n}-n+1 is automatically an integer). The divisors greater than 11 are
3, 9, 17, 51, 153.3,\ 9,\ 17,\ 51,\ 153.
 Step 3: That is 55 possible values of n.n. Correct answer: (2)
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