Question

If the ratio of the sum of the first \(\mathrm{m}\) terms and the first \(\mathrm{n}\) terms of an \(\mathrm{AP}\) is \(\mathrm{m}^{2}: \mathrm{n}^{2}\), the ratio of its \(\mathrm{mth}\) and \(\mathrm{nth}\) term will be (a) \(2 \mathrm{~m}-1: 2 \mathrm{n}-1\) (b) \(\mathrm{m}: \mathrm{n}\) (c) \(2 \mathrm{~m}+1: 2 \mathrm{n}+1\) (d) None of these

Short answer

Answer: (d) None of these.

Step-by-step solution

To find the sum of the first m and n terms of an AP, we can use the formula for the sum of an AP, which is given by: \(S_k = \frac{k}{2}(2a + (k-1)d),\) where \(S_k\) is the sum of the first k terms, a is the first term, d is the common difference, and k can be either m or n. In our case, we will have: \(S_m = \frac{m}{2}(2a + (m-1)d)\) and \(S_n = \frac{n}{2}(2a + (n-1)d)\). #Step 2: Substitute the given ratio and simplify#

According to the given information, the ratio of the sum of the first m terms to the sum of the first n terms of an AP is m^2 : n^2. Therefore, we can write this as a fraction: \(\frac{S_m}{S_n} = \frac{m^2}{n^2}.\) Now, substitute the expressions for \(S_m\) and \(S_n\) from Step 1: \(\frac{\frac{m}{2}(2a + (m-1)d)}{\frac{n}{2}(2a + (n-1)d)} = \frac{m^2}{n^2}.\) Simplify by canceling the \(\frac{1}{2}\) term on both sides: \(\frac{m(2a + (m-1)d)}{n(2a + (n-1)d)} = \frac{m^2}{n^2}.\) #Step 3: Find expressions for the mth and nth terms of the AP#

The kth term of an AP can be calculated by the formula: \(T_k = a + (k-1)d\), where \(T_k\) is the kth term, a is the first term, d is the difference, and k can be either m or n. So we have: \(T_m = a + (m-1)d\) and \(T_n = a + (n-1)d\). #Step 4: Calculate the ratio of the mth and nth terms of the AP#

We need to find the ratio of \(T_m\) and \(T_n,\) which can be written as: \(\frac{T_m}{T_n}.\) Using the expressions for the mth and nth terms from Step 3, we get: \(\frac{T_m}{T_n} = \frac{a + (m-1)d}{a + (n-1)d}\). Now, multiply both numerator and denominator of the obtained fraction by the expressions for \(S_m\) and \(S_n\) obtained from Step 2: \(\frac{T_m}{T_n} = \frac{(a + (m-1)d)\cdot m(2a + (m-1)d)}{(a + (n-1)d)\cdot n(2a + (n-1)d)}\). Comparing this expression to the given answer choices, we see that this ratio does not match any of the options provided. So, the correct answer is: (d) None of these.

Key concepts

Arithmetic Progression

An Arithmetic Progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. This property of having a fixed difference between consecutive terms is what defines an AP. For example, in the sequence 2, 4, 6, 8, the common difference is 2.

To further understand AP, imagine stacking blocks with equal spacings in between. Each block represents a term, and the gap between each pair of blocks is the common difference. If we label the first block as 'a' (the first term), then the next block would be 'a + d', where 'd' is the common difference. This pattern continues, allowing us to predict any term's value in the sequence based on its position.

Understanding APs is crucial in various real-life situations, such as calculating loan repayments, predicting savings account growth over time, or even understanding patterns within a set of data.

Sum of AP Formula

When we need to calculate the sum of a certain number of terms in an Arithmetic Progression (AP), we can utilize the sum of AP formula. The formula is \[ S_k = \frac{k}{2}(2a + (k-1)d) \]where \( S_k \) represents the sum of the first 'k' terms of the AP, 'a' is the first term, 'd' is the common difference, and 'k' can be any term number. You could picture this formula as a method for quickly adding up a long string of equally spaced numbers.

This formula is incredibly useful because it saves us from the time-consuming task of adding each term individually. It's particularly handy when dealing with large sequences, allowing us to efficiently calculate the sum without actually listing all the terms. In finance, for instance, this is analogous to finding the total amount of payments made over a period, given a fixed payment amount and frequency.

nth Term of AP

To find any specific term in an Arithmetic Progression, we use the formula for the nth term of AP, which is \[ T_n = a + (n-1)d \]where \( T_n \) is the nth term of the progression, 'a' is the first term, 'd' is the common difference, and 'n' represents the term number you wish to find. Consider this as locating a particular block in our earlier block-stacking analogy.

By simply knowing the starting point and the consistent step size, we can jump straight to any position and figure out the value of the block at that spot without having to step through every block in between. This kind of direct access is what makes an AP straightforward to work with and forms the basis of many algebraic methods and applications. It allows for the efficient planning and prediction of serial occurrences, such as scheduling events or understanding the progression of doses in a medication plan.