Ratio of terms

Algebra Level pending

The sum of the first ten terms of an arithmetic progression is four times the sum of its first five terms. Find the ratio of the first term to the third term?

1 : 5 1:5 1 : 2 1:2 1 : 3 1:3 1 : 6 1:6

Marvin Kalngan by Marvin Kalngan , 38, Philippines

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1 solution

Tom Engelsman
Apr 3, 2021

Let each term of this arithmetic series be represented by x + ( n − 1 ) y x + (n-1)y . We are given that:

Σ n = 1 10 x + ( n − 1 ) y = 4 [ Σ n = 1 5 x + ( n − 1 ) y ] ⇒ 10 x + 45 y = 4 ( 5 x + 10 y ) ⇒ y = 2 x \Sigma_{n=1}^{10} x + (n-1)y = 4[\Sigma_{n=1}^{5} x + (n-1)y] \Rightarrow 10x+ 45y = 4(5x+10y) \Rightarrow y = 2x .

If we desire the ratio of the first to the third term, then x x + 2 y = x x + 2 ( 2 x ) = 1 1 + 4 = 1 5 . \frac{x}{x+2y} = \frac{x}{x+2(2x)} = \frac{1}{1+4} = \boxed{\frac{1}{5}}.

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