Question

the sum of n terms of an AP is S=2n²+ 5n . find the 20th term of ​

Answer 1

$\huge\red{\boxed{\sf AnSwer}}$

Given :-

$\implies\sf\ S_n= 2n^2+5n$

To find :-

we have to find out the of 20th term of AP

Solution :-

As Given:

$\begin{gathered}\sf\ \ \ S_n= 2n^2+5n \\ \\ \\ \sf where\ (n)\ is\ the\ number\ of\ terms \\ \\ \sf \ By\ putting\ n= 1\\ \\ \\ \implies\sf\ \ S_{1}= 2(1)^2+5(1)\\ \\ \\ \implies\sf\ \ S_{1}= 2\times 1+5\\ \\ \\ \implies\sf\ S_{1}= 2+5\\ \\ \\ \implies\underline{\boxed{\sf{\purple{S_{1}=a_1= 7}}}}\\ \\ \\ \implies\sf\ S_2= 2(2)^2+5(2)\\ \\ \\ \implies\sf\ 2\times 4+10\\ \\ \\ \implies\underline{\boxed{\sf\ S_2= 18}}\\ \\ \\ \implies\sf\ \ a_1+a_2= 18\\ \\ \implies\sf\ a_2=18-7\\ \\\implies\sf\ a_2= 11\\ \\ \\ \implies\sf\ d= a_2-a_1\\ \\ \\ \implies\sf\ d= 11-7\\ \\ \underline{\boxed{\sf\ \ d= 4}}\\ \\ \\ \bf\ a_{20}= a+19d\\ \\ \\ \implies\sf\ a_{20}= 7+19\times 4\\ \\ \\ \implies\sf\ a_{20}= 7+76\\ \\ \\ \implies\underline{\boxed{\red{\sf a_{20}=532}}} \end{gathered}$

$\begin{gathered}\large\underline{\it{\bold \ Some \ formula\ realted\ AP}}\\ \\ \bullet\sf\ a_n= a+(n-1)d\\ \\ \bullet\sf\ S_n=\dfrac{n}{2}\big\{2a+(n-1)d\big\}\\ \\ \bf\ \ OR\\ \\ \bullet\sf\ S_n=\dfrac{n}{2}\big(a+\ell\big)\\ \\ \bullet\sf\ \ S_n-S_{n-1}= a_n\\ \\ \bullet\sf\ a_n-a_{n-1}= d\end{gathered}$

Answer 2

Given S

n

=2n

2

+5n

S

n−1

=2(n−1)

2

+5(n−1)

=2(n

2

−2n+1)+5n−5

=2n

2

−4n+2+5n−5

=2n

2

+n−3

Now, t

n

=S

n

−S

n−1

=2n

2

+5n−(2n

2

+n−3)

=2n

2

+5n−2n

2

−n+3

=4n+3