An infinite G.P. has first term 'x' and sum '5', then x belongs to (A) x

Q: Why can't $r$ be equal to 1 or -1?

Great question! If $r=1$, the sum is $a+a+a+...$, which diverges to infinity (or minus infinity). If $r=-1$, the sum is $a-a+a-a+...$, which oscillates and doesn't settle on a single finite value. For an infinite G.P. to have a finite sum, the terms must shrink towards zero, which only happens when $|r| < 1$.

Q: Why do we flip the inequality sign when multiplying by -1?

Think of it like a mirror reflection on the number line. If $2 > 1$, and you multiply both sides by $-1$, you get $-2$ and $-1$. Since $-1$ is to the right of $-2$ on the number line, $-1 > -2$. Multiplying by a negative number reverses the order, so the inequality sign must flip!

Q: What if $x=0$?

If $x=0$, the sum is $0+0+0+... = 0$. But the problem tells us the sum is $5$. Since $0$ is not $5$, $x$ cannot be $0$. That's why our final range is an open interval $(0, 10)$.

Question

An infinite G.P. has first term '$x$' and sum '$5$', then $x$ belongs to

Options:

(A) $x < -10$

(B) $-10 < x < 0$

(C) $0 < x < 10$

(D) $x > 10$

Sigma Percentile · Accepted Answer

### Analyzing the Setup Imagine standing before an infinite staircase where each step shrinks by a constant ratio. In the world of mathematics, this is an infinite geometric progression (G.P.). We are given an infinite G.P. with a first term $a = x$ and a total sum $S = 5$. Our goal is to determine the valid range of $x$. ### The Gatekeeper: Convergence Before we dive into the algebra, we must respect the "Gatekeeper" of infinite series: **convergence**. An infinite sum only exists if the common ratio $r$ satisfies the condition $|r| < 1$. If $|r| \ge 1$, the series diverges or oscillates. We utilize the fundamental formula for the sum of an infinite G.P.: $$S = \frac{a}{1 - r}$$ Given $S = 5$ and $a = x$, we substitute these values into the formula: $$5 = \frac{x}{1 - r}$$ ### The Algebraic Pivot Now, we manipulate the equation to isolate $r$ in terms of $x$. By cross-multiplying, we obtain: $$1 - r = \frac{x}{5}$$ Rearranging this expression, we find the common ratio: $$r = 1 - \frac{x}{5}$$ This result explicitly defines how the common ratio depends on our starting term $x$. ### The Inequality Trap To ensure the sum is exactly $5$, the common ratio $r$ must satisfy the convergence condition $-1 < r < 1$. Substituting our expression for $r$, we set up the following inequality: $$-1 < 1 - \frac{x}{5} < 1$$ First, subtract $1$ from all parts of the inequality: $$-2 < -\frac{x}{5} < 0$$ Next, we multiply the entire inequality by $-1$. Remember the golden rule: when multiplying or dividing an inequality by a negative number, the inequality signs must flip: $$2 > \frac{x}{5} > 0$$ Finally, multiply by $5$ to isolate $x$: $$10 > x > 0$$ ### The Final Revelation We have arrived at our destination. For an infinite G.P. to sum to $5$, the first term $x$ must lie in the range: **$$0 < x < 10$$** If $x$ were $10$ or greater, the common ratio would fail the convergence criteria. If $x$ were $0$ or less, the sum would not equal $5$. This result distills a complex, infinite concept into a clear, finite range.

Animated Solution for Mathematics - Sequence and Series: An infinite G.P. has first term ' $x$ ' and sum ' $5$ ', then $x$ belongs to

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Visualized Solution

Given Information

Sum Formula for Infinite G.P.

Substituting the Values

Convergence Condition

Expressing $r$ in terms of $x$

Isolating $r$

Setting up the Inequality

Solving the Inequality: Step 1

Solving the Inequality: Step 2

Solving the Inequality: Step 3

Final Range of $x$

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

Analyzing the Setup

The Gatekeeper

The Algebraic Pivot

The Inequality Trap

The Final Revelation

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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; } An infinite G.P. has first term 'x' and sum '5', then x belongs to