The paired sum follows an arithmetic progression too!

Algebra Level 1

The sum of the first two terms of an arithmetic progression is 2.
The sum of the next two terms is 4.
What is the sum of the subsequent two terms?

The answer is 6.

1 solution

Aaron Tsai
May 25, 2016

Relevant wiki: Arithmetic Progressions

Let $a$ be the first term of the sequence and $d$ be the common difference between 2 consecutive terms. We are given that

$\begin{cases}a+a+d=2\\a+2d+a+3d=4\end{cases}$

and we are asked to find $a+4d+a+5d$ , or $2a+9d$ . Solving the system of equations, we get $d=\dfrac{1}{2}\implies a=\dfrac{3}{4}$ .

So, $2a+9d=\dfrac{3}{2}+\dfrac{9}{2}=\boxed{6}$ .

Great! Can you solve this question without finding $a$ nor $d$ ?

And why is [ my title ] true?

Pi Han Goh - 5 years ago

An alternative solution:

The paired sums of an arithmetic sequence always form another arithmetic sequence. The sums are

$2a+d$ , $2a+5d$ , $2a+9d\ldots$

and that sequence has a common difference of $4d$ . From the first two terms we know that $4d=2$ . Since $2a+5d=4$ , it follows that $2a+9d=\boxed{6}$ .

Aaron Tsai - 5 years ago

The paired sum follows an arithmetic progression too!

The answer is 6.

1 solution

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