Question

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is \(6: 11\) and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

Short answer

The common difference of the A.P. is 9.

Step-by-step solution

Introduction to Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant, called the common difference, to the previous term. The n-th term of an AP is given by the formula: \(a_n = a_1 + (n - 1)d\), where \(a_1\) is the first term and \(d\) is the common difference.

Sum of Arithmetic Progression Terms

The sum of the first n terms of an AP can be calculated using the formula \(S_n = \frac{n}{2}(2a_1 + (n - 1)d)\), where \(S_n\) is the sum of the first n terms.

Setting Up the Ratio Equation

According to the given ratio of sums \(\frac{S_7}{S_{11}} = \frac{6}{11}\), we can express it using the sum formula for an AP as \(\frac{\frac{7}{2}(2a_1 + 6d)}{\frac{11}{2}(2a_1 + 10d)} = \frac{6}{11}\). Simplifying the equation gives us \(7 * (2a_1 + 6d) = 6 * (2a_1 + 10d)\).

Solve for the Common Difference

Expanding both sides of the equation from the previous step, we get \(14a_1 + 42d = 12a_1 + 60d\). Simplifying, we find \(2a_1 = 18d\), or \(a_1 = 9d\). Since the seventh term is given by \(a_7 = a_1 + 6d\), this means that \(a_7 = 9d + 6d = 15d\).

Finding the Value of the Seventh Term

Knowing that the seventh term lies between 130 and 140, we can set up inequalities \(130 < a_7 < 140\). Substituting \(a_7\) for \(15d\), we get \(130 < 15d < 140\). Dividing all parts of the inequality by 15 gives \(8.66 < d < 9.33\). Since d is a common difference in an AP consisting of natural numbers, d must also be natural. Therefore, \(d = 9\).

Key concepts

Common Difference

Understanding the 'common difference' is pivotal when working with arithmetic progressions (APs). It’s the consistent interval between consecutive terms in an AP, and it's this fixed increment that defines the progression. When we say a sequence is ‘arithmetic’, we mean it follows a simple pattern of adding or subtracting the same value—this value is the common difference, denoted by the symbol 'd'.

For instance, in the sequence 3, 6, 9, 12, ..., each term increases by 3; thus, the common difference is 3. It remains constant throughout the sequence, demonstrating that the pattern is linear. Being able to identify the common difference allows us to predict any term in the sequence and even find the sum of a set of its terms.

AP Sum Formula

The 'AP sum formula' is a straightforward way to calculate the sum of the first 'n' terms of an arithmetic sequence. The formula is expressed as:
\[S_n = \frac{n}{2}(2a_1 + (n - 1)d)\]
Where \(S_n\) represents the sum of the first 'n' terms, \(a_1\) is the first term, 'd' is the common difference, and 'n' is the number of terms to be added. This formula is derived from the sum of a linear series and is essential for efficiently finding the sum without having to add each term individually—a significant time-saver for longer sequences.

For example, if we want to find the sum of the first 5 terms of an arithmetic sequence starting with 2 and with a common difference of 3, we can quickly use the AP sum formula to arrive at the answer without manually adding 2 + 5 + 8 + 11 + 14.

Arithmetic Sequence

An 'arithmetic sequence' is a list of numbers with a specific order, where each number after the first is obtained by adding the common difference to its predecessor. You might recognize these sequences from various real-world phenomena such as evenly spaced time intervals or regularly scheduled payments.

Consider a clock ticking—each second increasing by the same interval is a physical representation of an arithmetic sequence in time. When studying these sequences in math, students must grasp that despite the variability of the first term and the common difference, the structure of the sequence remains constant. This uniformity underpins many concepts in arithmetic problems and helps in simplifying complex mathematical tasks.

Mathematical Inequalities

The realm of 'mathematical inequalities' plays a significant role when examining ranges in sequences, such as determining the common difference in an AP when given boundaries. Inequalities represent the relative size or order of two values, using symbols such as '<' for 'less than' and '>' for 'greater than'.

Applied to arithmetic progressions, inequalities can help us confine the possible values of a term or a common difference to a specific interval. For instance, if the common difference in an AP must be a natural number and we calculate that it should be greater than 8 but less than 9, we deduce that it must be exactly 9. Recognizing how to manipulate inequalities is crucial in math because it allows students to define the scope of possible solutions and to solve problems in a logical and systematic manner.