Understanding Infinite Geometric Progressions

An infinite geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).

The sum of an infinite geometric progression exists (converges) only if the absolute value of the common ratio is less than 1, i.e., $|r| < 1$ or $-1 < r < 1$.

The formula for the sum (S) of a convergent infinite GP is given by:

\( S = \frac{a}{1-r} \)

where \( a \) is the first term and \( r \) is the common ratio.

Applying the Infinite GP Sum Formula

In this problem, we are given:

• The first term \( a = x \)
• The sum of the infinite GP \( S = 5 \)

Using the sum formula, we have:

\( 5 = \frac{x}{1-r} \)

Finding the Common Ratio in Terms of x

We can rearrange the formula to express \( 1-r \) in terms of \( x \):

\( 1-r = \frac{x}{5} \)

Now, we can find the common ratio \( r \):

\( r = 1 - \frac{x}{5} \)

Applying the Convergence Condition

For the infinite GP to have a finite sum, the common ratio \( r \) must satisfy the condition \( -1 < r < 1 \).

Substitute the expression for \( r \) into the inequality:

\( -1 < 1 - \frac{x}{5} < 1 \)

Solving the Inequality for x

We need to solve this compound inequality for \( x \).

First, subtract 1 from all parts of the inequality:

\( -1 - 1 < 1 - \frac{x}{5} - 1 < 1 - 1 \)

\( -2 < - \frac{x}{5} < 0 \)

Next, multiply all parts of the inequality by -5. Remember that when multiplying or dividing an inequality by a negative number, the direction of the inequality signs must be reversed.

\( (-2) \times (-5) > \left( - \frac{x}{5} \right) \times (-5) > 0 \times (-5) \)

\( 10 > x > 0 \)

This inequality can be read as "10 is greater than x, and x is greater than 0", which is the same as:

\( 0 < x < 10 \)

Conclusion

For an infinite GP with first term \( x \) and sum 5 to exist, the value of \( x \) must be between 0 and 10 (exclusive).

Revision Table: Key Concepts

Additional Information: Convergence of GP

The convergence of a geometric progression is determined solely by its common ratio, \( r \).

• If \( |r| < 1 \), the terms get progressively smaller, approaching zero. The sum of infinitely many terms converges to a finite value given by \( \frac{a}{1-r} \).
• If \( |r| \ge 1 \), the terms do not approach zero (or they oscillate without approaching zero). The sum of infinitely many terms diverges, meaning it does not approach a finite value.
• If \( r = 1 \), all terms are equal to the first term \( a \). The sum goes to infinity if \( a \ne 0 \).
• If \( r = -1 \), the terms alternate between \( a \) and \( -a \). The sum oscillates between \( a \) and 0 (or \( a \) and \( -a \)), and does not converge.

Understanding the condition \( |r| < 1 \) is crucial for problems involving sums of infinite geometric sequences.