If the sum of the first $n$ terms of an A.P. is $4n – n^2$, what is the first term? What is the sum of first two terms? What is the second term? Similarly, find the third, the tenth and the $n$th terms.
Given:
The sum of first $n$ terms of an A.P. is $4n-n^{2}$.
To do:
We have to find the first term, sum of first two terms, second term, third term, tenth term and the $n$th term.
Solution:
$S_{n} =4n-n^{2}$
For $n=1,\ S_{1} =4\times 1 -1^2=4-1=3$
Therefore, first term $a=3$
For $n=2,\ S_{2} =4\times 2-2^{2}=8-4=4$
$\therefore$ Second term of the A.P.$=S_{2} -S_{1}$
$=4-3$
$=1$
Common difference of the A.P., $d=$second term $-$ first term
$=1-3=-2$
We know that,
$a_{n}=a+(n-1)d$
$a_{2}=a+d=3+(-2)=3-2=1$
$a_3=a+2d=3+2(-2)=3-4=-1$
$a_{10}=a+(10-1)d=3+9(-2)=3-18=-15$
$a_n=a+(n-1)d=3+(n-1)(-2)$
$=3-2n+2$
$=5-2n$
Therefore, the first term is $3$, the second term is $1$, the third term is $-1$, the tenth term is $-15$, $n$th term is $5-2n$ and the sum of the first two terms is $4$.
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