Animated Solution for Mathematics - Sequence and Series: If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:

Let us define our Arithmetic Progression (A.P.) with the first term $a$ and the common difference $d$. The problem directs our attention to the 2nd, 5th, and 9th terms.

Using the standard formula for the $n$-th term of an A.P., $a_n=a+(n-1)d$, we express these terms as: $a_2=a+d$ $a_5=a+4d$ $a_9=a+8d$

These expressions serve as the fundamental building blocks for our solution.

Bridge

The problem states that the terms $a_2,a_5,a_9$ form a Geometric Progression (G.P.). If three numbers $x,y,z$ are in a G.P., the ratio of consecutive terms must be equal, leading to the condition $y^2=xz$.

Applying this property to our specific terms, we obtain the following equation:

Substituting our A.P. expressions into this relation, we arrive at:

We now perform the algebraic expansion. On the left side, using the identity $(x+y{)}^2=x^2+2xy+y^2$, we obtain:

On the right side, expanding the product of the binomials $(a+d)(a+8d)$ yields:

Equating the two sides, we have:

The $a^2$ term appears on both sides and cancels out completely. We are left with the simplified equation:

We rearrange the terms to group the variables:

The condition that the A.P. is non-constant implies $d eq 0$. This allows us to divide both sides by $d$ without loss of generality, yielding the relationship:

The common ratio $r$ of a G.P. is defined as the ratio of any term to its predecessor. Thus:

Substituting our derived relation $a=8d$ into this expression, we get:

The $d$ terms cancel out, leaving us with the final result:

$$r=\frac{4}{3}$$