Question

If the sum of the first n terms of an A.P. is 4n – n², which 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 nth terms.

Answer 1

Given the sum of n terms of an A.P -

Sn = 4n -n²

The first term of the A.P = S1 = 4-1 = 3 = a

=> a = 3

The sum of the first terms of the A.P = S2

= 4

=> a + a + d = 4

=> 2a + d = 4

=> d = -2

The common difference of the arithmetic progression is equal to -2.

The second term of the A.P = a + d = 1

The third term of the A.P = a + 2d = -1

The tenth term of the A.P = a + 9d = -15

The nth term of the given A.P = 3 + (n-1)×-2

Answer 2

( i ) The First term is 3

( i i ) The Second term is 4

( i i i )The sum of first two terms is 7

(i v ) The Third term is 5

( v ) The tenth term is 12

( v i ) The nth terms is (2 + n)

Step-by-step explanation:

Given as :

The sum of first n terms of an A.P = 4 n - n²

i.e   $S_n$  = $\dfrac{n}{2}$ [ a + (n - 1) d ]  = 4 n - n²

Or, $S_n$  =  4 n - n²

( i ) The First term

$t_1$ = 4 × 1 - 1²

Or,   $t_1$ = 4 - 1 = 3

So, The first term = $t_1$ = 3

( i i ) The Second term

 $t_2$ = 4 × 2 - 2²

Or,       $t_2$ = 8 - 4 = 4

So, The second term = $t_2$ = 4

Now,

( i i i ) The sum of first two terms

$t_1$  +   $t_2$

Or,  $t_1$  +   $t_2$  = 3 + 4  = 7

∴ sum of first two terms = 7

Again

the first term = $t_1$ = a = 3

common difference = d = $t_2$  - $t_1$  =  4 - 3

i.e    d = 1

(i v ) The Third term

$t_3$ = a + (n - 1) d

i.e  $t_3$ = 3 + (3 - 1) 1

or,  $t_3$ = 3 + 2 = 5

So, The third term = 5

( v ) The tenth term

$t_1_0$ = a + (n - 1) d

i.e  $t_1_0$ = 3 + (10 - 1) 1

or,  $t_1_0$ = 3 + 9 = 12

So, The tenth term = 12

( v i ) The nth terms

∵ $t_n$  = 3 + (n - 1) 1

or, $t_n$ = 3 + n - 1

∴   $t_n$ = 2 + n

So, The nth term = 2 + n

Hence,

( i ) The First term is 3

( i i ) The Second term is 4

( i i i )The sum of first two terms is 7

(i v ) The Third term is 5

( v ) The tenth term is 12

( v i ) The nth terms is (2 + n)              Answer