Question

If t(n) represents the $n^{th}$ term of an AP, then which of the following is always true?

• A. t(1) + t(20) = t(19) + t(2)
• B. t(2) + t(18) = t(4) + t(16)
• C. t(1) + t(20) = t(21)
• D. t(9) + t(10) = t(19)

Answer 1

Answer: (A), (B)

The correct options are
A t(1) + t(20) = t(19) + t(2)
B t(2) + t(18) = t(4) + t(16)

We know that the sum of the first term and the last term is equal to the sum of the second term and the second last term and so on.

Therefore,
t(1) + t(n) = t(2) + t(n-1) = t(3) + t(n-2) and so on.

Observe that in all the cases, the sum of the numbers in the brackets (which determines which term it is) is always equal to (n+1).

So, $(1+n)=(2+(n-1\left)\right)=(3+(n-2\left)\right)=......$

So, we can come up with the following observation:
In an AP where t(n) represents the $n^{th}$ term, if t(a) + t(b) = t(x) + t(y), then a + b = x + y