# The fourth term of an arithmetic progression is \[10\] and the eleventh term of it exceeds three times the fourth term by $1$. Find the sum of the first $20$ terms of the progression.

Last updated date: 27th Mar 2023

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Answer

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Hint : Find first term and common difference to find the sum.

As we know that the nth term of an A.P. is denoted as ,

${a_n} = a + (n - 1)d$

From the question we can say that,

$

{a_4} = a + (4 - 1)d = a + 3d = 10\,\,\,\,...(i) \\

{a_{11}} = a + (11 - 1)d = a + 10d\,\,\,\, \\

$

It is given that,

$

{a_{11}} = 3{a_4} + 1 \\

a + 10d = 3a + 9d + 1 \\

2a - d + 1 = 0\,\,\,\,\,\,\,\,\,......(ii) \\

$

On multiplying the equation $(ii)$ by 3 we get,

$6a - 3d + 3 = 0\,\,\,\,\,\,......(iii)$

Solving equation $(i)$ & $(iii)$we get,

$

a = 1 \\

\& \\

d = 3 \\

$

We know sum on n terms of an A.P. can be written as

$

{S_n} = \frac{n}{2}(2a + (n - 1)d) \\

{S_{20}} = \frac{{20}}{2}(2(1) + 19 \times 3) \\

{S_{20}} = 590 \\

$

Hence, the sum of 20 terms of the series in 590.

Note :- In these types of questions of A.P. we have to first obtain an equation from the given data then solve the equation to get the unknowns like first term & common difference, after finding the unknowns, obtain the sum by using the formula of sum of an A.P.

As we know that the nth term of an A.P. is denoted as ,

${a_n} = a + (n - 1)d$

From the question we can say that,

$

{a_4} = a + (4 - 1)d = a + 3d = 10\,\,\,\,...(i) \\

{a_{11}} = a + (11 - 1)d = a + 10d\,\,\,\, \\

$

It is given that,

$

{a_{11}} = 3{a_4} + 1 \\

a + 10d = 3a + 9d + 1 \\

2a - d + 1 = 0\,\,\,\,\,\,\,\,\,......(ii) \\

$

On multiplying the equation $(ii)$ by 3 we get,

$6a - 3d + 3 = 0\,\,\,\,\,\,......(iii)$

Solving equation $(i)$ & $(iii)$we get,

$

a = 1 \\

\& \\

d = 3 \\

$

We know sum on n terms of an A.P. can be written as

$

{S_n} = \frac{n}{2}(2a + (n - 1)d) \\

{S_{20}} = \frac{{20}}{2}(2(1) + 19 \times 3) \\

{S_{20}} = 590 \\

$

Hence, the sum of 20 terms of the series in 590.

Note :- In these types of questions of A.P. we have to first obtain an equation from the given data then solve the equation to get the unknowns like first term & common difference, after finding the unknowns, obtain the sum by using the formula of sum of an A.P.

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