Different A.P. 's are constructed with first term 100 ,last term 199 ,And integral common differ

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8. Sequences and Series

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Different $A.P.$'s are constructed with the first term $100$,the 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.

A

$54$

B

$52$

C

$56$

D

$53$

(JEE MAIN-2022)

Solution

$1^{\text {st }} \text { term }=100=a$

Last term $=199=\ell$

If $3$ term

$a, a+d, a+2 d$

$a _{ a }=\ell= a +( n -1) d$

$d _{ i }=\frac{\ell- a }{ n – l }$

$n \rightarrow$ number of terms

$n =3, d _{1}=\frac{199-100}{2}$

$=\frac{99}{2} \notin I$

$n =4, d _{2}=\frac{99}{3}=33 \in I$

$n =10, d _{3}=\frac{99}{9}=11 \in I$

$n =12, d _{4}=\frac{99}{11}=9 \in I$

$\therefore \sum d _{ i }=33+11+9=53$

Standard 11

Mathematics

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