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# In an A.P. the sum of first ten terms is$210$ and the difference between the first and last term is$36$.Find the first term in the A.P.$A.{\text{ }}2 \\ B.{\text{ }}3 \\ C.{\text{ }}4 \\ D.{\text{ }}5 \\$

Last updated date: 31st Mar 2023
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Hint- Obtain the equations using given information and use known formulas of Arithmetic Progression , clearly sum of first n terms of AP formula will be used here.
Let${S_n}$ denote the sum of $n$ terms.
We know that,
${S_n} = \dfrac{n}{2}\left( {a + {a_n}} \right)$, where $a$ is the first term and ${a_n}$ is the last term.
Now, we have given that the sum of the first ten terms is $210$.
Therefore, the number of terms is $10$.
$\Rightarrow {S_n} = \dfrac{{10}}{2}\left( {a + {a_n}} \right) \\ \Rightarrow 210 = 5\left( {a + {a_n}} \right) \\ \Rightarrow \dfrac{{210}}{5} = \left( {a + {a_n}} \right) \\ \Rightarrow 42 = a + {a_n} - - - - \left( i \right) \\$
Also, the difference between the first and last term is $36$.
$36 = {a_n} - a - - - - \left( {ii} \right)$
Solving $\left( i \right)$ and $\left( {ii} \right)$ equations simultaneously we get,
${a_n} = 39$
Putting the value of ${a_n}$ in equation $\left( i \right)$ we get,
$a = 3$.
Hence the first term is $3.$

Note- Whenever we face such types of questions the key concept is that we should write what is given to us. Then write the formula of sum of series in an AP and then put values in the formula and thus we get the answer.