91Ó°ÊÓ

A geometric series is a sequence where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general form is:

• a, ar, ar^2, ar^3, \textellipsis

where 'a' is the first term, and 'r' is the common ratio. For example, if you have a series 2, 4, 8, 16, \textellipsis, here 'a' is 2 and 'r' is 2.
To calculate the sum of the first 'n' terms of a geometric series, the formula is: \[ S_n = \frac{a(r^n - 1)}{r - 1} \] This formula holds true as long as r ≠ 1. Apply the provided formula by plugging in the values for a, r, and n, and you will get the sum for part (a) of the exercise, \( \sum_{n=1}^{20}(1.1)^{n} \). Hence, we have a = 1.1, r = 1.1. The calculation is as follows: \[ S_{20} = \frac{1.1(1.1^{20} - 1)}{0.1} \] This simplifies to approximately 62.9.

An arithmetic series is a sequence where the difference between consecutive terms is constant. This difference is called the ‘common difference’. Its general form is:

• a, a + d, a + 2d, a + 3d, \textellipsis

where 'a' is the first term and 'd' is the common difference. For example, in the sequence 3, 6, 9, 12, \textellipsis, 'a' is 3 and 'd' is 3.
The sum of the first 'n' terms of an arithmetic series is given by the formula: \[ S_n = \frac{n}{2} (2a + (n - 1)d) \] However, in this exercise, for part (b), the terms are in an arithmetic progression in a slightly different way due to logarithmic properties. First, transform the series using: \( \log_{10}(1.1^n) = n \log_{10}(1.1) \). Then factor out \( \log_{10}(1.1) \) and sum the remaining arithmetic series: \[ \sum_{n=1}^{20} n \log_{10}(1.1) \] Compute the sum of the first 20 positive integers (an arithmetic series) using: \[ \sum_{n=1}^{20} n = \frac{20(20+1)}{2} = 210 \]. Hence, the series sum simplifies to: \( 0.0414 \times 210 = 8.70 \).

Logarithms are mathematical operations that are the inverse of exponentiation. They have several useful properties that help simplify complex expressions. Here are some key properties:

• \( \log_{b}(xy) = \log_{b}(x) + \log_{b}(y) \) (Product Rule)
• \( \log_{b}(\frac{x}{y}) = \log_{b}(x) - \log_{b}(y) \) (Quotient Rule)
• \( \log_{b}(x^n) = n \log_{b}(x) \) (Power Rule)

In this exercise, the power rule is used for part (b), converting parts of the logarithmic series: \( \log_{10}(1.1^n) = n \log_{10}(1.1) \). This makes the computation easier, allowing you to factor out \( \log_{10}(1.1) \) and sum the series of integers. Knowing these properties not only helps in simplifying the problems but also in understanding the structure and behavior of logarithmic functions.