Which of the following relations is always true for an arithmetic progression, where $r$ is a positive integer and all indices are valid?

Question

Answer and explanation

Correct answer: $a_n=\frac{a_{n-r}+a_{n+r}}{2}$

In an AP, $a_k=a+(k-1)d$. Thus $a_{n-r}=a_n-rd$ and $a_{n+r}=a_n+rd$, so their average is $a_n$. Exam tip: remember this for equally spaced terms.

What is the correct answer to this question?

$a_n=\frac{a_{n-r}+a_{n+r}}{2}$

Why is this the correct answer?

In an AP, $a_k=a+(k-1)d$. Thus $a_{n-r}=a_n-rd$ and $a_{n+r}=a_n+rd$, so their average is $a_n$. Exam tip: remember this for equally spaced terms.

Which subject and chapter does this question cover?

This is a Class 10 Mathematics question. Chapter: Arithmetic Progressions (AP). Topic: Finding the $n$th term of an AP.