# Which term of the A.P 21, 42, 63, 84..…………. is 420?

Last updated date: 18th Mar 2023

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Answer

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Hint – In this question use the formula of ${n^{th}}$ term of an A.P which is given as ${a_n} = {a_1} + \left( {n - 1} \right)d$, so use this A.P property to reach the answer.

Given A.P is

21, 42, 63, 84…………….

So, the first term $\left( {{a_1}} \right)$ of this A.P $ = 21$

Common difference (d) of this A.P $ = \left( {42 - 21} \right) = \left( {63 - 42} \right) = 21$

So according to formula of ${n^{th}}$ term of an A.P which is,

${a_n} = {a_1} + \left( {n - 1} \right)d.............\left( 1 \right)$, where n is number of terms.

Now we have to find out which term of this A.P is 420.

$ \Rightarrow {a_n} = 420$

Now from equation (1)

$

420 = 21 + \left( {n - 1} \right)\left( {21} \right) \\

\Rightarrow 21\left( {n - 1} \right) = 420 - 21 = 399 \\

\Rightarrow n - 1 = \frac{{399}}{{21}} = 19 \\

\Rightarrow n = 19 + 1 = 20 \\

$

Therefore 420 is the ${20^{th}}$ term of this A.P.

Note – whenever we face such types of problems the key concept we have to remember is that always recall all the basic formulas of A.P which is stated above, then first find out the first term and common difference of given A.P and substitute these values in the above formula and calculate which term of the A.P is 420.

Given A.P is

21, 42, 63, 84…………….

So, the first term $\left( {{a_1}} \right)$ of this A.P $ = 21$

Common difference (d) of this A.P $ = \left( {42 - 21} \right) = \left( {63 - 42} \right) = 21$

So according to formula of ${n^{th}}$ term of an A.P which is,

${a_n} = {a_1} + \left( {n - 1} \right)d.............\left( 1 \right)$, where n is number of terms.

Now we have to find out which term of this A.P is 420.

$ \Rightarrow {a_n} = 420$

Now from equation (1)

$

420 = 21 + \left( {n - 1} \right)\left( {21} \right) \\

\Rightarrow 21\left( {n - 1} \right) = 420 - 21 = 399 \\

\Rightarrow n - 1 = \frac{{399}}{{21}} = 19 \\

\Rightarrow n = 19 + 1 = 20 \\

$

Therefore 420 is the ${20^{th}}$ term of this A.P.

Note – whenever we face such types of problems the key concept we have to remember is that always recall all the basic formulas of A.P which is stated above, then first find out the first term and common difference of given A.P and substitute these values in the above formula and calculate which term of the A.P is 420.

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