Question

Sequence and series

Answer 1

\[\LARGE a_1,a_2,a_3\] are in AP then \[\LARGE a_p,a_q,a_r \]
are in AP if p,q,r in:
AP,GP,HP,n/t

Answer 2

@sirm3d

Answer 3

consider the AP \[\Large a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, a_{10}, \dots\] and the subset \[\Large b_1=a_1, b_2=a_3, b_3=a_5, b_4 = a_7,\dots\] check the terms if they form an AP

Answer 4

that is, compare the differences \(b_2-b_1, b_3-b_2, b_4-b_3,\dots\)

Answer 5

AP?

Answer 6

is the ans?

Answer 7

If you want to be certain, the consider that AP series has the form: \[
a_n = a_0 + (n-1)d
\]Now what if you have another AP: \[
b_n = b_0 + (n-1)d
\]Plug \(b_n\) into the \(n\) in the equation for \(a_n\) and see if it becomes for sure an AP.

Answer 8

thanks!

Answer 9

You can also try: \[\large
b_n = b_0r^{n-1}
\]To test GP and so on.