Question

Find (i) the 22 nd term of the progression \(4,9,14,19 \ldots \ldots \ldots\) (ii) the first negative term of the progression \(19,18 \frac{1}{5}, 17 \frac{2}{5} \ldots \ldots\) (iii) the 11 th term of the series \(5+10+20+\ldots \ldots . .\) (iv) the GP if 4 th and 9 th terms are 54 and 13122 respectively.

Short answer

\(T_{22} = 4 + (21)(5)\) \(T_{22} = 4 + 105\) \(T_{22} = 109\) The 22nd term of the arithmetic progression is 109.

Step-by-step solution

Problem (i): Find the 22nd term of the AP

To find the 22nd term of the arithmetic progression \(4,9,14,19,\ldots\), we need to identify the common difference (d) and use the formula for the nth term of an AP: \(T_n =a + (n - 1)d\) where \(T_n\) is the nth term, "a" is the first term, and "n" is the number of terms. Here, the common difference (d) can be calculated as the difference between any two consecutive terms: \(d = 9 - 4 = 5\). Now, substitute the values into the formula: \(T_{22} = 4 + (22 - 1)(5)\).

Key concepts

AP formula

Understanding the arithmetic progression (AP) formula is essential for solving problems involving a sequence of numbers with a constant difference between them. The formula for the nth term of an arithmetic progression, as stated in the step by step solution, is given by:

\[T_n = a + (n - 1)d\]

Where:\[T_n\] represents the nth term of the AP, \(a\) is the first term in the sequence, \(n\) is the term number, and \(d\) is the common difference between each pair of consecutive terms. The common difference can be determined by subtracting the first term from the second term in the sequence. Once the common difference is known, the AP formula allows you to calculate any term in the sequence simply by plugging the values for \(a\), \(n\), and \(d\) into the formula. This formula is a quick and elegant method to navigate through arithmetic sequences without necessarily listing every single term.

Common difference

At the heart of every arithmetic progression lies the 'common difference'. This value, often denoted as \(d\), is the uniform interval between each term in the series. It's calculated by finding the difference between two successive terms.

To find the common difference, you can select any term in the series (after the first) and subtract the immediate preceding term. For instance, if you have an AP with terms \(a, a+d, a+2d, \textellipsis\), then the common difference is \(d = (a+d) - a\). In the given solution, the common difference for the AP \(4, 9, 14, 19, \textellipsis\) is determined by subtracting 4 from 9, which gives us a common difference of 5. Recognizing the common difference is a key step before applying the AP formula, as it is necessary for calculating any term within the sequence.

Geometric progression

A geometric progression (GP), unlike an arithmetic progression, 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'. Problems involving geometric progressions require you to understand this concept of a common ratio, typically denoted as \(r\).

For example, if you have the series \(b, br, br^2, \textellipsis\), then \(r = \frac{br}{b}\) is your common ratio. This ratio remains consistent throughout the series. Once you have the first term and the common ratio, you can find any term in the GP using the formula:

GP nth Term Formula

\[G_n = b \times r^{(n-1)}\]
Where \(G_n\) is the nth term, \(b\) is the first term, and \(n\) is the term number. In more complex scenarios, like problem (iv) in the original exercise, finding the common ratio may require working backward from the terms provided. It is crucial to understand this relationship as it forms the backbone for solving various problems related to geometric series.

Series

In mathematics, a series is the sum of the terms of a sequence. Sequences come in different forms, including arithmetic and geometric, each with its way of determining the sum. For an arithmetic series, this sum is the result of adding up all the terms in the arithmetic progression. The formula for the sum of the first \(n\) terms of an AP is given as:

Arithmetic Series Sum Formula

\[S_n = \frac{n}{2}(2a + (n - 1)d)\]
For a geometric series, the sum depends on the common ratio. If the common ratio is greater than 1, the series has a tendency to grow larger with each term; if it is less than 1, the series converges. The formula for the sum of a geometric series when the common ratio (\(r\)) is not equal to 1 is:

Geometric Series Sum Formula

\[S_n = b \frac{1 - r^n}{1 - r}\]
Where \(S_n\) is the sum of the first \(n\) terms, and other symbols retain their previous meanings. In problem (iii) of the original exercise, you are asked to find the 11th term of the series. This calls for the application of the geometric progression formula since the terms are each twice the previous term. Summation of a series is a fundamental concept often applied in various fields such as economics, computer science, and engineering, where understanding the total effect of a sequence of events or numbers is crucial.