# Consider the A.P. $2,5,8,11,.......,302$ .Show that twice of the middle term of the above A.P. is equal to the sum of its first and last term.

Last updated date: 29th Mar 2023

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Answer

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Hint: Use ${n^{th}}$ term formula ${a_n} = a + \left( {n - 1} \right)d$

It is given $2,5,8,11,.......,302$ is an A.P. with first term $a = 2$ and common difference $d = \left( {II} \right)term - \left( I \right)term = 5 - 2 = 3$ .Let there be $n$ terms in the given A.P.

Then, ${n^{th}}$ term $\left( {{a_n}} \right) = 302$

Formula of ${n^{th}}$term is ${a_n} = a + \left( {n - 1} \right)d$

$

\Rightarrow a + \left( {n - 1} \right)d = 302 \\

\Rightarrow 2 + 3\left( {n - 1} \right) = 302 \\

\Rightarrow 2 + 3n - 3 = 302 \\

\Rightarrow 3n = 303 \\

\Rightarrow n = 101 \\

\\

$

Clearly, $n$ is odd. Therefore , ${\left( {\dfrac{{n + 1}}{2}} \right)^{th}}$ i.e. ${51^{st}}$ term is the middle term.

Now, middle term $ = {a_{51}} = a + 50d = 2 + 50 \times 3 = 152$

Sum of first term and last term $ = 2 + 302 = 304$

Twice of middle term $ = 2 \times 152 = 304$

So, Twice of the middle term of the above A.P. is equal to the sum of its first and last term.

Note: Whenever we come across these types of problems first we have to find the first term and common difference of given A.P., then using ${n^{th}}$ term formula find the number of terms in an A.P. If number of terms is odd then for middle term use ${\left( {\dfrac{{n + 1}}{2}} \right)^{th}}$

After finding the middle term again use the ${n^{th}}$ term formula to find the value of the middle term.

It is given $2,5,8,11,.......,302$ is an A.P. with first term $a = 2$ and common difference $d = \left( {II} \right)term - \left( I \right)term = 5 - 2 = 3$ .Let there be $n$ terms in the given A.P.

Then, ${n^{th}}$ term $\left( {{a_n}} \right) = 302$

Formula of ${n^{th}}$term is ${a_n} = a + \left( {n - 1} \right)d$

$

\Rightarrow a + \left( {n - 1} \right)d = 302 \\

\Rightarrow 2 + 3\left( {n - 1} \right) = 302 \\

\Rightarrow 2 + 3n - 3 = 302 \\

\Rightarrow 3n = 303 \\

\Rightarrow n = 101 \\

\\

$

Clearly, $n$ is odd. Therefore , ${\left( {\dfrac{{n + 1}}{2}} \right)^{th}}$ i.e. ${51^{st}}$ term is the middle term.

Now, middle term $ = {a_{51}} = a + 50d = 2 + 50 \times 3 = 152$

Sum of first term and last term $ = 2 + 302 = 304$

Twice of middle term $ = 2 \times 152 = 304$

So, Twice of the middle term of the above A.P. is equal to the sum of its first and last term.

Note: Whenever we come across these types of problems first we have to find the first term and common difference of given A.P., then using ${n^{th}}$ term formula find the number of terms in an A.P. If number of terms is odd then for middle term use ${\left( {\dfrac{{n + 1}}{2}} \right)^{th}}$

After finding the middle term again use the ${n^{th}}$ term formula to find the value of the middle term.

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