Question

If $a_1,a_2,a_3,...(a_i\ne a_j\ne 0)$ and $b_1,b_2,b_3,...(b_i\ne b_j\ne 0)$ are two different A.P., then which of the following is/are always correct?

• A. $a_1,a_3,a_5,a_7,...$ are in A.P.
• B. $b_2,b_6,b_{10},b_{14},...$ are in A.P.
• C. $a_1+b_1,a_2+b_2,a_3+b_3,...$ are in A.P.
• D. $\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},...$ are in A.P.

Answer 1

Answer: (A), (B), (C)

The correct options are
A $a_1,a_3,a_5,a_7,...$ are in A.P.
B $b_2,b_6,b_{10},b_{14},...$ are in A.P.
C $a_1+b_1,a_2+b_2,a_3+b_3,...$ are in A.P.

We know that, if the terms of an A.P. are chosen at regular intervals, then they also form an A.P.
$\Rightarrow a_1,a_3,a_5,a_7,...$ are in A.P.
$\Rightarrow b_2,b_6,b_{10},b_{14},...$ are in A.P.

We also know that, the sequence obtained by adding or subtracting the corresponding terms of the two A.P.s is also an A.P,
$\Rightarrow a_1+b_1,a_2+b_2,a_3+b_3,...$ are in A.P.