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प्रश्न
The sum of the first n terms of an A.P. is 4n2 + 2n. Find the nth term of this A.P ?
उत्तर १
\[\text{The sum of n terms of an A . P . is given by}\ S_n = \frac{n}{2}\left( a + T_n \right), \text{where a} = 1^{st} \text{term and}\ T_n = n^{th} \text{term}.\]
\[\text{So, we have}: \]
\[4 n^2 + 2n = \frac{n}{2}\left( a + T_n \right)\]
\[ \Rightarrow n\left( 4n + 2 \right) = \frac{n}{2}\left( a + T_n \right)\]
\[ \Rightarrow 8n + 4 = a + T_n . . . (1)\]
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:
\[ \Rightarrow T_n = 8n - 2\]
उत्तर २
\[\text{The sum of n terms of an A . P . is given by}\ S_n = \frac{n}{2}\left( a + T_n \right), \text{where a} = 1^{st} \text{term and}\ T_n = n^{th} \text{term}.\]
\[\text{So, we have}: \]
\[4 n^2 + 2n = \frac{n}{2}\left( a + T_n \right)\]
\[ \Rightarrow n\left( 4n + 2 \right) = \frac{n}{2}\left( a + T_n \right)\]
\[ \Rightarrow 8n + 4 = a + T_n . . . (1)\]
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:
\[ \Rightarrow T_n = 8n - 2\]
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