Question

The numbers 1,5 and 25 can be three terms (not necessarily consecutive) of

Answer 1

We see that 25/5=5 and 5/1=1

so the common ratio is 5

The given terms form a Geometric Progression with first term 1 and common ratio 5

Answer 2

Answer:

The numbers 1,5 and 25 can be three terms (not necessarily consecutive) of one or many numbers of Arithmetic and Geometric Progressions.

Step-by-step explanation:

To check the Arithmetic Progression:

Let us assume that given number series 1,5,25 are I-th, J-th and Q-th terms in the AP series whose distance is equal to d.

Therefore,  

(J – I)d = 5-1

(Q-I)d = 25 – 1

(J – I) / 4 = (Q-I) /24

(J – I) / 1 = (Q-I) / 6 = x

So, J - I = x

J = I + x

Q = I + 6x

Here x is a natural number.

So, the given three numbers can be terms in one or many number of Arithmetic Progressions.

To check for Geometric Progression:

Let us assume that the given number series 1,5,25 are I-th, J-th and Q-th terms in the GP series whose ratio is equal to r.

So,

r^( J – I) = 5;

r^(Q-I) = 25 = 5^2

Therefore,

Q-I = 2(J – I)

I – 2J + Q = 0

This relation has many solutions.

So, the given three numbers can be terms in one or many number of Geometric Progressions.