Question

If the sum of the first n terms of an AP is4n-n^2, what is the first term (remember the first term is S1 ) ? What is the first term? What are the 2nd term,3rd term , 10th term and nth terms.

Answer 1

Solution:

Given: $S_n=4n-n^2$

To find: $a_1,a_2,a_3,a_{10},a_n$=?

We know that,
$S_1=a_1$

$S_1=4(1)-(1)^2$
$a_1=4-1$
    [tex]a =3 [/tex] ..(i)

We know that $S_2=a_1+a_2$
                        
$S_2=4(2)-(2)^2$
$S_2=8-4$
$S_2=4$
$a_1+a_2=4$
$3+a_2=4$

We know that $a_n=a+(n-1)d$

$3+(a+(2-1)d)=4$
$a+d=4-3$
$3+d=1$
$d=1-3$
$d=-2$ ..(ii)

$a_2=a+(2-1)d$
$a_2=3+(-2)$
$a_2=1$

$a_3=a+(3-1)d$
$a_3=3+(2)(-2)$
$a_3=3-4$
$a_3=-1$

[tex]a_{10}=a+(10-1)d [/tex]
$a_{10}=3+(9)(-2)$
$a_{10}=3-18$
$a_{10}=-15$

$a_n=a+(n-1)d$
$a_n=3+(n-1)(-2)$
$a_n=3-2n+2$
$a_n=5-2n$

                 Hope it helps