Question

Q.12
If the sum of the first n terms of an AP is 4n-n?, find 2nd and
nth term.​

Answer 1

Answer:

Second term = 1

$\sf{n^{th}}$ term = 5 - 2n

Step-by-step explanation:

The sum of first n terms of an AP is 4n - n².

→ $\sf{S_n\:=\:4n\:-\:n^2}$

Now, take n = 1

→ $\sf{S_1\:=\:4(1)\:-\:(1)^2}$

= $\sf{4\:-\:1}$

= $\sf{3}$

Now, take n = 2

→ $\sf{S_2\:=\:4(2)\:-\:(2)^2}$

= $\sf{8\:-\:4}$

= $\sf{4}$

We have to find the second term of an AP i.e. $\sf{a_2}$

Sum of first two terms i.e. Sum of $\sf{S_1}$ and $\sf{S_2}$ is 4.

→ $\sf{a_2\:=\:4\:-\:3}$

= $\sf{a_2\:=\:1}$

First term = 3 and Second term = 1

$\therefore$ Common difference = Second term - First term

= $\sf{1\: - \:3}$

= $\sf{-\:2}$

So, $\sf{n^{th}}$ term of an AP :-

→ $\sf{a_n\:=\:a\:+\:(n\:-\:1)d}$

From the above calculations we have -

• a = first term = 3
• d = common difference = -2

Substitute the known values in the above formula.

= $\sf{3\:+\:(n\:-\:1)(-2)}$

= $\sf{3\:-\:2n\:+\:2}$

= $\sf{5\:-\:2n}$

Answer 2

$\bold{\large{\underline{\underline{\sf{StEp\:by\:stEp\:explanation:}}}}}$

The sum of first n terms of an AP is 4n - n².

→ $\huge\tt\green{S_n\:=\:4n\:-\:n^2}$

Now, put n = 1

◆ $\tt{S_1\:=\:4(1)\:-\:(1)^2}$

= 4 - 1

= 3

Now, put n = 2

◆ $\tt{S_2\:=\:4(2)\:-\:(2)^2}$

= 8−4

= 4

We have to find the second term of an AP i.e. $\tt{a_2}$

Sum of first two terms i.e. Sum of $\tt{S_1}$ and $\tt{S_2}$ is 4.

$\tt{a_2\:=\:4\:-\:3}$

=4−3

= 1

hence ,

First term = 3

Second term = 1

Common difference = Second term - First term

d = 1- 3 = -2

So, $\tt{n^{th}}$ term of an AP :-

=a+(n−1)d

From the above calculations we have -

a = first term = 3

d = common difference = -2

So by putting the known values in the formula.

= $\tt {3\:+\:(n\:-\:1)(-2)}$

= $\tt{3\:-\:2n\:+\:2}$

= $\tt{5\:-\:2n}$