Question

If the sum of the first n terms of an AP is $4n-n^2,$ what is the first term (that is SI) ? What is the sum of first two terms ? What is the second term ? Similarly. find the $3^{rd}$ and ${10}^{th}$ and the $n^{th}$ terms.
If $S_n=4n-n^2,$ then
i) $S_1=a=?$
ii) $S_2=?$
iii) $a_2=?$
iv) $a_3=?$
v) $a_{10}=?$
vi) $a_n=?$

Answer 1

(i) $S_n=4n-n^2$

$S_1=4\left(1\right)-1^2$

$=4-1$

$S_1=3$

$\therefore S_1=a=3.$

(ii) $S_2=4n-n^2$

$S_2=4\left(2\right)-2^2$

$=8-4$

$S_2=4$

$\therefore S_2=4$

(iii) We have $S_2=a_1+a_2=4$

$=3+a_2=4$

$\therefore a_2=4-3$

$\therefore a_2=1$

(iv) $S_n=4n-n^2$

$S_3=4\left(3\right)-3^2$

$=12-9$

$S_3=3$

$a_1+a_2+a_3=3$

$3+1+a_3=3$

$4+a_3=3$

$\therefore a_3=3-4$

$\therefore a_3=-1$

(v) $d=a_3-a_2=-1-1=-2$

$a_{10}=a+9d$

$=3+9(-2)$

$=3-18$

$a_{10}=-15$

$\therefore a_{10}=-15$

(vi) $a_n=a+(n-1)d$

$=3+(n-1)(-2)$

$=3-2n+2$

$\therefore a_n=5-2n$

$\therefore a_n=5-2n$