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The Sum of First Seven Terms of an A.P. is 182. If Its 4th and the 17th Terms Are in the Ratio 1 : 5, Find the A.P. - Mathematics

Sum

The sum of first seven terms of an A.P. is 182. If its 4th and the 17th terms are in the ratio 1 : 5, find the A.P.

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Solution

Let a be the first term and d be the common difference.
We know that, sum of first n terms = Sn = \[\frac{n}{2}\][2a + (n − 1)d]

According to the question,

\[S_7 = 182\]
\[ \Rightarrow \frac{7}{2}\left[ 2a + \left( 7 - 1 \right)d \right] = 182\]
\[ \Rightarrow \frac{1}{2}\left( 2a + 6d \right) = 26\]
\[ \Rightarrow a + 3d = 26\]
\[ \Rightarrow a = 26 - 3d . . . . (1)\]

Also,

\[\frac{a_4}{a_{17}} = \frac{1}{5}\]

\[ \Rightarrow \frac{a + (4 - 1)d}{a + (17 - 1)d} = \frac{1}{5}\]

\[ \Rightarrow \frac{a + 3d}{a + 16d} = \frac{1}{5}\]

\[ \Rightarrow 5(a + 3d) = a + 16d\]

\[ \Rightarrow 5a + 15d = a + 16d\]

\[ \Rightarrow 5a - a = 16d - 15d\]

\[ \Rightarrow 4a = d . . . . (2)\]

On substituting (2) in (1), we get

\[a = 26 - 3\left( 4a \right)\]
\[ \Rightarrow a = 26 - 12a\]
\[ \Rightarrow 12a + a = 26\]
\[ \Rightarrow 13a = 26\]
\[ \Rightarrow a = 2\]
\[ \Rightarrow d = 4 \times 2 \left[ \text{ From }  \left( 2 \right) \right]\]
\[ \Rightarrow d = 8\]

Thus, the A.P. is 2, 10, 18, 26, ..... .

 
 
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APPEARS IN

RD Sharma Class 10 Maths
Chapter 5 Arithmetic Progression
Exercise 5.6 | Q 31 | Page 52
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