The sum of the first ten terms of an…

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The sum of the first ten terms of an Arithmetic Progression (A.P.) is 130. If the fifth term is 3 times the first term, find the:

Answer

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Let the first term be \(a\) and the common difference \(d\).

Sum of 10 terms: \(S_{10} = \tfrac{10}{2}(2a + 9d) = 130 \Rightarrow 2a + 9d = 26\).

Fifth term is three times the first: \(a + 4d = 3a \Rightarrow 4d = 2a \Rightarrow a = 2d\).

Substitute: \(2(2d) + 9d = 26 \Rightarrow 13d = 26 \Rightarrow d = 2\), so \(a = 4\).

Common difference \(= 2\); first term \(= 4\).

Number of terms when the last term is 28:

\[a + (n - 1)d = 28 \Rightarrow 4 + 2(n - 1) = 28 \Rightarrow n - 1 = 12 \Rightarrow n = 13\]

The sum of the first ten terms of an Arithmetic Progression (A.P.) is 130. If the fifth term is 3 times the first term, find the: common difference; first t...