If the sum of the first 𝑛 terms of an A.P. is then find its term $\frac{1}{2}$[3𝑛² + 7𝑛], 𝑛^𝑡ℎ and its 20 term.

We have been given the sum of the first 𝑛 terms of an A.P. as and $\frac{1}{2}$ [3𝑛 2 + 7𝑛], 

we need to find the 𝑛^th and the 20^th term.

We know that formula to find sum of 𝑛 terms, $S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

Where 𝑆_n is the sum 𝑛 of terms, 𝑎 is the first term, 𝑛 is the term to be found and is the 𝑑 common difference of the A.P.

$\Rightarrow S_n=\frac{1}{2}\left[3n^2+7n\right]$

Putting 𝑛 = 1 will give us the first term of the A.P

$\begin{array}{r}\Rightarrow S_1=\frac{1}{2}\left[3(1{)}^2+7(1)\right]\\ \Rightarrow S_1=\frac{1}{2}\times 10\\ \Rightarrow S_1=5\end{array}$

To find the sum of first two terms, we can put n = 2

$\begin{array}{r}\Rightarrow S_2=\frac{1}{2}\left[3(2{)}^2+7(2)\right]\\ \Rightarrow S_2=\frac{1}{2}\left[26\right]\\ \Rightarrow S_2=13\end{array}$

Now, first term of the 𝐴. 𝑃 = 𝑆₁ 

Second term of the 𝐴. 𝑃. = 𝑆𝑢𝑚 𝑜𝑓 𝑓𝑖𝑟𝑠𝑡 𝑡𝑤𝑜 𝑡𝑒𝑟𝑚𝑠 − 𝐹𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚

⇒Second term of the 𝐴. 𝑃. = S₂ - S₁

= 13 − 5 

= 8 

Common difference of an 𝐴𝑃 is the difference between its two consecutive terms.

⇒ Common difference; 𝑑 = 8 − (5) 

⇒ 𝑑 = 3

Any 𝐴. 𝑃 can be expressed using 𝑎 and 𝑑 as : 𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛 Therefore, the 𝐴. 𝑃. is 5, 5 + 3, 5 + 2 × 3, 5 + 3 × 3 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛 

Hence, the 𝐴. 𝑃 = 5, 8, 11,..

Now, the 𝑛^th term can be found out using the formula 

𝑛^𝑡ℎ 𝑡𝑒𝑟𝑚 = 𝑎 + (𝑛 − 1)d

where 𝑎 is the first term of the 𝐴. 𝑃, 𝑑 is its common difference, and 𝑛 is the number of terms,

𝑛^𝑡ℎ 𝑡𝑒𝑟𝑚 = 5 + (𝑛 − 1) × 3

⇒ 𝑛^𝑡ℎ 𝑡𝑒𝑟𝑚 = 3𝑛 + 2

The 20^𝑡ℎ term can be found using the same formula. 

20^𝑡ℎ 𝑡𝑒𝑟𝑚 = 𝑎 + (𝑛 − 1)𝑑

= 5 + (20 − 1) × 3 

= 5 + 57 

⇒ 20^𝑡ℎ 𝑡𝑒𝑟𝑚 = 62

Hence, the 𝑛^th term = 3n + 1 and 20^th the term of the given A.P is 62.