Class 11 Math Algebra Sequences and series

The sum of $n$ terms of an arithmetic series is $S_{n}=2n−n_{2}$. Find the first term and the common difference.

(a)

$a=2;d=2$

(b)

$a=0;d=1$

(c)

$a=1;d=−2$

(d)

$a=−1;d=2$

Correct answer: (c)

Solution: The sum of n terms formula is given by,

$S_{n}=2n [2a+(n−1)d]$ ........... (1)

Given :- $S_{n}=2n−n_{2}$ ......... (2)

Comparing equation (1) & (2), we get

$2n (2a+(n−1)d)=2n−n_{2}$

$2an+n_{2}d−nd=4n−2n_{2}$

$(2a−d)n+dn_{2}=4n−2n_{2}$ .......... (3)

On comparing factors of $n_{2}$ in equation (3)

$d=−2$

On comparing factors of $n$ in equation (3), we have

$2a−d=4$

$2a=4+d$

$2a=4−2$

$2a=2$

$a=1$

Therefore, the first term, $a=1$ and the common difference, $d=−2$.

$S_{n}=2n [2a+(n−1)d]$ ........... (1)

Given :- $S_{n}=2n−n_{2}$ ......... (2)

Comparing equation (1) & (2), we get

$2n (2a+(n−1)d)=2n−n_{2}$

$2an+n_{2}d−nd=4n−2n_{2}$

$(2a−d)n+dn_{2}=4n−2n_{2}$ .......... (3)

On comparing factors of $n_{2}$ in equation (3)

$d=−2$

On comparing factors of $n$ in equation (3), we have

$2a−d=4$

$2a=4+d$

$2a=4−2$

$2a=2$

$a=1$

Therefore, the first term, $a=1$ and the common difference, $d=−2$.

Similar topics

determinants

matrices

binomial theorem

complex numbers and quadratic equations

sequences and series

determinants

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binomial theorem

complex numbers and quadratic equations

sequences and series

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