Different A.P.'s are constructed with first term 100, last term 199, and integral common d…

Question

Different A.P.'s are constructed with first term $100$, last term $199$, and integral common differences. The sum of the common differences of all such A.P.'s having at least $3$ terms and at most $33$ terms is \_\_\_\_\_\_\_\_.

Answer: 53

Accepted Answer

If there are $i+1$ terms ($2\le i\le32$), the common difference is $d_i = \dfrac{99}{i+1}$, which must be an integer --- so $i+1$ must divide $99$ and lie in $[3,33]$. The valid divisors are $3, 9, 11$, giving $d=33, 11, 9$.

$33+11+9 = 53.$

Solutions · 1