Question

If the sum of the first $n$ terms of an AP is $4n-n^2$, what is the first term (that is $S_1$) What is the sum of first two terms? What is the second term? Similarly, find the $3rd$, the $10th$ and the nth terms.

Answer 1

$s_n=4n-n^2$

$T_n=s_n-s_{n-1}$

$T_n=4n-n^2-4(n-1)+(n-1{)}^2$

$T_n=4n-n^2-4n+4+n^2-2n+1$

$T_n=-2n+5$

$T_1=-2\times 1+5=3$

$T_2=-2\times 2+5=1$

$T_3=-2\times 3+5=-1$

$T_{10}=-2\times 10+5=-15$

sum of first two terms$=T_1+T_2=3+1=4$

second term$=T_2=1$

third term$=T_3=-1$

tenth term$=t_{10}=-15$