Question

If the first term of an AP is 4 and the sum of the first four terms is equal to the one-third of the sum of the next four terms. What will be the sum of first 20 terms of the given AP?

Answer 1

Answer:

If the sum of first four terms of an AP is 40 and that of first 14 terms is 280. Find the sum of its first n terms.

Step-by-step explanation:

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Answer 2

Given: First term of an AP is 4, sum of the first four terms is equal to one-third of the sum of the next four terms.

To find: the sum of first 20 terms of the given AP.

Solution:

Find the common difference of the AP.

Understand that, $\[{a_1} = 4\]$ and

$\[{a_1} + {a_2} + {a_3} + {a_4} = \frac{1}{3}\left( {{a_5} + {a_6} + {a_7} + {a_8}} \right)\]$

$\[ \Rightarrow a + \left( {a + d} \right) + \left( {a + 2d} \right) + \left( {a + 3d} \right) = \frac{1}{3}\left[ {\left( {a + 4d} \right) + \left( {a + 5d} \right) + \left( {a + 6d} \right) + \left( {a + 7d} \right)} \right]\]$

$\[ \Rightarrow 4a + 6d = \frac{1}{3}\left( {4a + 22d} \right)\]$

$\[\begin{array}{l} \Rightarrow 12a + 18d = 4a + 22d\\ \Rightarrow 12a - 4a = 22d - 18d\\ \Rightarrow 8a = 4d\end{array}\]$

$\[ \Rightarrow d = \frac{8}{4}a\]$

$\[\begin{array}{l} \Rightarrow d = 2a\\ \Rightarrow d = 2 \times 4\\ \Rightarrow d = 8\end{array}\]$

Therefore, the common difference is 8.

Find the sum of first 20 terms of the AP.

$\[{S_n} = \frac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]\]$

$\[ \Rightarrow {S_{20}} = \frac{{20}}{2}\left[ {2 \times 4 + \left( {20 - 1} \right)8} \right]\]$

$\[\begin{array}{l} \Rightarrow {S_{20}} = 10\left[ {8 + \left( {19} \right)8} \right]\\ \Rightarrow {S_{20}} = 10\left[ {8 + 162} \right]\\ \Rightarrow {S_{20}} = 10\left[ {170} \right]\\ \Rightarrow {S_{20}} = 1700\end{array}\]$

Hence, the first 20 terms of an AP is 1700.