Advertisements
Advertisements
Question
The sum of the first n terms of an A.P. is 4n2 + 2n. Find the nth term of this A.P ?
Solution 1
\[\text{The sum of n terms of an A . P . is given by}\ S_n = \frac{n}{2}\left( a + T_n \right), \text{where a} = 1^{st} \text{term and}\ T_n = n^{th} \text{term}.\]
\[\text{So, we have}: \]
\[4 n^2 + 2n = \frac{n}{2}\left( a + T_n \right)\]
\[ \Rightarrow n\left( 4n + 2 \right) = \frac{n}{2}\left( a + T_n \right)\]
\[ \Rightarrow 8n + 4 = a + T_n . . . (1)\]
Now, we have:
S1=a⇒a=412+21=6" data-mce-style="display: inline; font-style: normal; font-weight: 400; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: #212121; font-family: Roboto, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-style: initial; text-decoration-color: initial; position: relative;" data-mce-tabindex="0">\[S_1 = a\]
\[ \Rightarrow a = \left( 4 \left( 1 \right)^2 + 2\left( 1 \right) \right) = 6\]
Putting the value of a in equation (1), we get:
\[ \Rightarrow T_n = 8n - 2\]
Solution 2
\[\text{The sum of n terms of an A . P . is given by}\ S_n = \frac{n}{2}\left( a + T_n \right), \text{where a} = 1^{st} \text{term and}\ T_n = n^{th} \text{term}.\]
\[\text{So, we have}: \]
\[4 n^2 + 2n = \frac{n}{2}\left( a + T_n \right)\]
\[ \Rightarrow n\left( 4n + 2 \right) = \frac{n}{2}\left( a + T_n \right)\]
\[ \Rightarrow 8n + 4 = a + T_n . . . (1)\]
Now, we have:
S1=a⇒a=412+21=6" data-mce-style="display: inline; font-style: normal; font-weight: 400; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: #212121; font-family: Roboto, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-style: initial; text-decoration-color: initial; position: relative;" data-mce-tabindex="0">\[S_1 = a\]
\[ \Rightarrow a = \left( 4 \left( 1 \right)^2 + 2\left( 1 \right) \right) = 6\]
Putting the value of a in equation (1), we get:
\[ \Rightarrow T_n = 8n - 2\]
APPEARS IN
RELATED QUESTIONS
In the following APs, find the missing term in the boxes:
`-4, square, square, square, square, 6`
The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the A.P.
Write the first five terms of the following sequences whose nth terms are:
`a_n = (2n - 3)/6`
The sum of 4th and 8th terms of an A.P. is 24 and the sum of 6th and 10th terms is 44. Find the A.P.
If the nth term of a progression is (4n – 10) show that it is an AP. Find its
(i) first term, (ii) common difference (iii) 16 the term.
A child puts one five-rupee coin of her saving in the piggy bank on the first day. She increases her saving by one five-rupee coin daily. If the piggy bank can hold 190 coins of five rupees in all, find the number of days she can continue to put the five-rupee coins into it and find the total money she saved.
Write your views on the habit of saving.
37th term of the AP: `sqrtx, 3sqrtx, 5sqrtx, ....` is ______.
Justify whether it is true to say that the following are the nth terms of an AP.
3n2 + 5
Fill in the blank in the following table, given that a is the first term, d the common difference, and an nth term of the AP:
| a | d | n | an |
| ______ | -3 | 18 | -5 |
In an A.P., the first term is 12 and the common difference is 6. If the last term of the A.P. is 252, then find its middle term.
