Problem 5 - Technical Mathematics with Calculus

Question

Find the sum of the infinitely many terms of each GP. $$144,72,36,18, \dots$$

Step-by-Step Solution

Answer

The sum of the infinite geometric progression is 288.

Step 1: Identify the first term and common ratio

First, identify the first term, denoted as 'a', and the common ratio of the geometric progression (GP), denoted as 'r'. In the sequence given, the first term is 'a = 144' and each subsequent term is half of the previous one, meaning the common ratio 'r' is '0.5' or '1/2'.

Step 2: Apply the formula for the sum of an infinite GP

For an infinite GP, the sum to infinity is given by the formula \(S = \frac{a}{1 - r}\), where 'S' is the sum to infinity, only if the common ratio '|r| < 1'. Plug in the values of 'a' and 'r' as identified in Step 1.

Step 3: Compute the sum

Substitute 'a = 144' and 'r = \frac{1}{2}' into the formula to get \(S = \frac{144}{1 - \frac{1}{2}} = \frac{144}{\frac{1}{2}} = 144 \times 2 = 288\). Therefore, the sum of the GP to infinity is 288.

Key Concepts

Geometric Progression

A 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. This type of sequence has various applications in fields such as finance, biology, physics, and computer science, where it can describe growth or decay, like interest rates or radioactive decay rates, respectively.

For instance, consider the sequence 144, 72, 36, 18, ... Here, each term is created by multiplying the previous term by 1/2. The sequence progresses in a manner where the ratio between consecutive terms remains constant, which is the defining characteristic of a GP.

Common Ratio

The common ratio in a geometric progression is key to determining the pattern of the sequence. It's the factor by which we multiply each term to get to the next one. In a formula, if the first term of a GP is denoted by 'a' and the common ratio by 'r', the nth term ('T_n') is given by:

\[ T_n = a \times r^{(n-1)} \]
For the given example with the first term as 144 and the subsequent term as 72, we can calculate the common ratio by dividing any term by its preceding term, resulting in a common ratio (r) of 0.5. It's crucial that the common ratio is not 0 because it would make all subsequent terms 0, thereby making it non-geometric.

Sum to Infinity Formula

The sum to infinity of a geometric progression is an important concept especially when dealing with series that go on indefinitely. This sum only converges to a finite value if the absolute value of the common ratio is less than one (\(|r| < 1\)). The formula to find the sum is:

\[ S = \frac{a}{1 - r} \]
where 'S' is the sum to infinity, 'a' is the first term, and 'r' is the common ratio. Referring to our exercise, where 'a' is 144 and 'r' is 0.5, the sum to infinity would be calculated as \[ S = \frac{144}{1 - 0.5} = 288 \].

This formula allows us to evaluate the sum of infinitely many terms of a GP quite easily, as long as the sequence meets the condition for convergence. Remember, this incredible concept lets us sum an endless number of terms into a neat and finite number!