Question

The fourth term of an A.P. is 0. Prove that the ${25^{th}}$ term of the A.P. is three times its ${11^{th}}$ term.

Answer 1

Hint: We will first write the formula for the ${n^{th}}$ term of an A.P. Now, consider to find the ${25^{th}}$ term and modify it such that it is some linear combination of the ${11^{th}}$ and ${4^{th}}$ term to get the required result.

Complete step-by-step answer:
Let us first get to know something about A.P.
An arithmetic progression (AP), also called an arithmetic sequence, is a sequence of numbers which differ from each other by a common difference. For example, the sequence 2, 4, 6, 8, ……..… is an arithmetic sequence with the common difference 2.
If we have an A.P. with the first terms as a and common difference being d, then its ${n^{th}}$ term is given by the formula:- ${a_n} = a + (n - 1)d$ …………….(1)
Now, since we are given that the fourth term is zero.
This can be written as:-
$ \Rightarrow {a_4} = 0$
Now, using (1), this can be written as follows:-
$ \Rightarrow {a_4} = a + 3d = 0$
Hence, we have: $a + 3d = 0$
Taking the $3d$ to the RHS, we have:- $a = - 3d$ ………(2)
Now, consider the ${25^{th}}$ term of this A.P:
\[ \Rightarrow {a_{25}} = a + 24d\] (Using (1))
Now, putting in (2) in this, we will get:-
\[ \Rightarrow {a_{25}} = a + 24d = - 3d + 24d\]
Simplifying the RHS to get as follows:-
\[ \Rightarrow {a_{25}} = 21d\] …………..(3)
Now, let us consider the ${11^{th}}$ term of this A.P:
\[ \Rightarrow {a_{11}} = a + 10d\] (Using (1))
Now, putting in (2) in this, we will get:-
\[ \Rightarrow {a_{11}} = a + 10d = - 3d + 10d\]
Simplifying the RHS to get as follows:-
\[ \Rightarrow {a_{11}} = 7d\] …………..(4)
Now, if we multiply the equation (4) with 3, we will get:-
\[ \Rightarrow 3{a_{11}} = 21d\]
Now, comparing this to (3), we will get as follows:-
\[ \Rightarrow {a_{25}} = 3{a_{11}}\].
Hence proved.

Note: The students must note that they need as much of equations as much unknown variables they have. Like, in this question, we formed two equations for two unknown variables a and d.
Fun Facts:-If a, b, c are in AP, then 2b = a + c.
If each term of an AP is increased, decreased, multiplied, or divided by a constant non-zero number, then the resulting sequence is also in AP.