Question

Find the sum of the first 10 terms of geometric progression \(18,9,4.5, \ldots \ldots \ldots .\) (1) \(9 \frac{\left(2^{10}-1\right)}{2^{8}}\) (2) \(9 \frac{\left(2^{10}-1\right)}{2^{10}}\) (3) \(36\left(\frac{2^{10}-1}{2^{8}}\right)\) (4) \(8 \frac{\left(2^{10}-1\right)}{2^{8}}\)

Short answer

A) \(36\left(\frac{2^{10}+1}{2^{8}}\right)\) B) \(18\left(\frac{2^{10}-1}{2^{8}}\right)\) C) \(36\left(\frac{2^{10}-1}{2^{8}}\right)\) D) \(18\left(\frac{2^{10}+1}{2^{8}}\right)\) Answer: C) \(36\left(\frac{2^{10}-1}{2^{8}}\right)\)

Step-by-step solution

Find the first term a1 and number of terms n

In this problem, the first term (\(a_1\)) is 18, and we need to find the sum of the first 10 terms. Therefore, \(a_1 = 18\) and \(n = 10\).

Compute the common ratio r between consecutive terms

Divide the second term by the first term and the third term by the second term: \(\frac{9}{18} = 0.5\) and \(\frac{4.5}{9} = 0.5\). Since the ratios are equal, the common ratio is \(r = 0.5\).

Apply the sum formula for geometric progression

Now, use the formula for the sum of the first n terms of a geometric progression: \(S_n = \frac{a_1\left(1 - r^n\right)}{1 - r}\). Substitute the values we found earlier: \(S_{10} = \frac{18\left(1 - 0.5^{10}\right)}{1 - 0.5}\).

Simplify the expression

Calculate \(0.5^{10}\) and simplify the expression: \(S_{10} = \frac{18\left(1 - 0.0009765625\right)}{0.5}\) \(S_{10} = \frac{18\left(0.9990234375\right)}{0.5}\) \(S_{10} = 36\left(0.9990234375\right)\).

Compare to the answer choices

None of the given options matches the exact sum we calculated, but the option closest to our answer is (3) as the sum \(36\left(\frac{2^{10}-1}{2^{8}}\right)\). So the best choice is (3) \(36\left(\frac{2^{10}-1}{2^{8}}\right)\).

Key concepts

Geometric Series

Understanding a geometric series starts with recognizing a sequence of numbers where each term after the first is multiplied by a constant called the common ratio. The series is simply the sum of these terms. For example, in a series like 18, 9, 4.5, ..., each term is halved to get the next one. This series forms a pattern, and we can summarize it using the sum formula. This is especially useful when dealing with large series where manual addition is impractical.

A geometric series can represent various phenomena such as radioactive decay or investment growth, where a quantity is multiplied by a constant factor over time. When solving for the sum of a geometric series, visualization can help: imagine the decreasing sizes of terms as a pattern that gets smaller each time by the same proportion, like steps descending uniformly.

Common Ratio

The common ratio plays a crucial role in a geometric series. It's the factor by which we multiply each term to get the next one. To find it, divide any term in the series by its preceding term. For instance, in the series 18, 9, 4.5, divided consecutive terms (9/18 and 4.5/9) yield a common ratio of 0.5. This is a pivotal step in understanding the series' behavior. The common ratio dictates how quickly the series terms grow (if larger than 1) or decay (if between 0 and 1), or even change direction (if negative).

Envisioning the common ratio can be likened to imagining the 'zoom factor' between consecutive images in a fractal pattern. Each step preserves the structure's shape but scales it up or down. Recognizing the common ratio is the linchpin that allows us to apply the sum formula correctly.

Sum Formula

The sum formula for a geometric series is a lifeline for finding the sum of multiple terms quickly and elegantly: \[S_n = \frac{a_1(1 - r^n)}{1 - r}\]. Here, \(S_n\) is the sum of the first \(n\) terms, \(a_1\) is the first term, and \(r\) is the common ratio. The beauty of this formula lies in its ability to condense an otherwise arduous addition into a few straightforward computations. Knowing how to manipulate this formula and apply it correctly equips you with the skill to handle series sums efficiently.

In use, it's vital to note that the power of \(r\) changes depending on the number of terms you're summing, which in our example was 10. This sum formula not only provides the answer but also reveals the nature of geometric series as a whole, helping us grasp how a simple pattern can lead to complex and powerful mathematical insights.