Question

If the sum of the first n terms of an $A.P$. 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

If Sum of first $n$ terms of an AP

$S_n=4n-n^2$

Then, the first term will be when $n=1$ in the above expression

$S_1=a=4\left(1\right)-(1{)}^2=3$

Sum of first two terms,

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

So, the second term will be

$a_2=S_2-S_1=4-3=1$

Similarly

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

$\Rightarrow a_3=S_3-S_2=3-4=-1$

$n=9,S_9=4\left(9\right)-9^2=-45$

$n=10,s_{10}=4\left(10\right)-(10{)}^2=-60$

$\Rightarrow a_{10}=S_{10}-S_9=-60-(-45)=-15$

$a_n=S_n-S_{(}n-1)$

$\Rightarrow a_n=4n-n^2-\left(4\right(n-1)-(n-1{)}^2)$

$\Rightarrow a_n=5-2n$