An algebra problem by Paola Ramírez

Algebra Level 3

The first four terms of an arithmetic sequence are a + b , a − b , a b a+b,a-b,ab and a b \frac{a}{b} .

If the 5th term can be written as m n \dfrac mn , where m m and n n are coprime positive integers, find the value of m + n m+n .

This problem is not original.


The answer is 163.

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Paola Ramírez by Paola Ramírez , 24, Mexico

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1 solution

Paola Ramírez
Jul 30, 2015

First, find the difference between one term and the netx which is a constant, substracting two terms.

( a + b ) − ( a − b ) = − 2 b (a+b)-(a-b)=\boxed{-2b}

Then construct a system of equations in order to find the values of a a and b b using the constant term.

a − 3 b = a b a-3b=ab

a − 5 b = a b a-5b=\frac{a}{b}

Solving the system,get a = − 9 8 a=-\frac{9}{8} and b = − 3 5 ⇒ b=-\frac{3}{5} \Rightarrow the 5 t h 5^{th} term is a b − 2 b = − 9 8 − 3 5 − 2 ( − 3 5 ) = 123 40 ∴ m + n = 163 \frac{a}{b}-2b=\frac{-\frac{9}{8}}{-\frac{3}{5}}-2(-\frac{3}{5})=\frac{123}{40}\therefore m+n=\boxed{163}

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Yes same I did

Rakshit Joshi - 5 years, 8 months ago

I used The very same method too

Shreyash Rai - 5 years, 6 months ago

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