Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP i

Q: Wait, why do we need $|r| < 1$?

That is a brilliant question! In an infinite Geometric Progression, if the common ratio $r$ is greater than or equal to $1$ (or less than or equal to $-1$), the terms do not get smaller. Instead, they grow or stay the same, meaning the sum would blow up to infinity. The formula $S_{\infty} = \frac{a}{1-r}$ is only valid when the series converges, which strictly requires $|r| < 1$.

Q: Can I solve for $a$ in the first equation and substitute it into the second?

Absolutely! That is a perfectly valid algebraic path. You would get $a = 20(1-r)$, and then substitute $a^2 = 400(1-r)^2$ into the second equation. It leads to the same result, but the method of squaring the first equation and dividing is often faster and less prone to calculation errors. Both ways show you are thinking critically!

Q: What if the sum of squares was smaller than the square of the sum?

That is a great intuition check. If you ever find yourself in a situation where the math seems to contradict the physical reality of the problem, pause and re-check your equations. In this specific case, the relationship between the sum of the GP and the sum of the squares is fixed by the properties of $r$. If the numbers given were impossible, the math would lead to a value of $r$ that violates $|r| < 1$.

Question

Sum of infinite number of terms of GP is $20$ and sum of their square is $100$. The common ratio of GP is

Options:

(A) 5

(B) 3/5

(C) 8/5

(D) 1/5

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing at the edge of an infinite staircase. Each step you take is smaller than the last, shrinking by a constant factor $r$. This is the essence of a **Geometric Progression (GP)**.

We start with a first term $a$ and a common ratio $r$. The sequence unfolds as $a, ar, ar^2, ar^3, \dots$ and continues forever.

The beauty of this sequence lies in its convergence. As long as the common ratio $r$ is between $-1$ and $1$, the sum of all these infinite terms settles into a finite, elegant value:

$$S_{\infty} = \frac{a}{1-r}$$

We are told this sum is $20$, so we write our first anchor point:

$$\frac{a}{1-r} = 20$$

### The Squared Transformation

Now, let us perform a transformation. What happens if we square every single term in our sequence? The new sequence becomes $a^2, (ar)^2, (ar^2)^2, \dots$, which simplifies to $a^2, a^2r^2, a^2r^4, \dots$.

Look closely—this is still a **Geometric Progression**! Its first term is now $A = a^2$, and its common ratio is $R = r^2$.

The problem tells us the sum of this new, squared sequence is $100$. Using our sum formula again, we get:

$$\frac{a^2}{1-r^2} = 100$$

### The Algebraic Symphony

We now have a system of two equations:
(1) $\frac{a}{1-r} = 20$
(2) $\frac{a^2}{1-r^2} = 100$

Our goal is to isolate $r$. The most efficient way to do this is to eliminate $a$. If we square the first equation, we get:

$$\frac{a^2}{(1-r)^2} = 400$$

By dividing this new squared equation by the second equation, the $a^2$ terms vanish into thin air:

$$\frac{\frac{a^2}{(1-r)^2}}{\frac{a^2}{1-r^2}} = \frac{400}{100}$$

This simplifies to:

$$\frac{1-r^2}{(1-r)^2} = 4$$

### The Final Resolution

We are almost at the finish line. Recall the difference of squares identity: $1-r^2 = (1-r)(1+r)$. Substituting this into our equation gives us:

$$\frac{(1-r)(1+r)}{(1-r)^2} = 4$$

We can cancel the common factor $(1-r)$ from the numerator and denominator, leaving us with:

$$\frac{1+r}{1-r} = 4$$

Cross-multiplying, we get $1+r = 4(1-r)$, which expands to $1+r = 4-4r$. Grouping the $r$ terms gives $5r = 3$, and finally:

**$$r = \frac{3}{5}$$**

This value, $0.6$, is less than $1$, confirming our series converges perfectly. You have successfully navigated the infinite!

Animated Solution for Mathematics - Sequence and Series: Sum of infinite number of terms of GP is $20$ and sum of their square is $100$ . The common ratio of GP is

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Visualized Solution

Defining the Infinite GP

Sum of the Infinite GP

The Squared Sequence

Sum of the Squared GP

Strategy: Squaring the First Equation

Dividing to Eliminate $a$

Simplifying the Complex Fraction

Expanding the Numerator

Canceling Common Factors

Cross-Multiplying

Solving for $r$

Final Conclusion

The Sigma Insight: Geometric Progression (G.P.)

Analyzing the Setup

The Squared Transformation

The Algebraic Symphony

The Final Resolution

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Animated Solution for Mathematics - Sequence and Series: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is