Solving Arithmetic Progression Problems Finding M And The Sum

This article delves into solving a problem related to arithmetic progressions (A.P.). We are given the first three terms of an A.P. as (m+1), (4m-2), and (6m-3), and we know the last term is 18. Our task is to determine the value of 'm' and calculate the sum of all the terms in this A.P. This involves understanding the fundamental properties of A.P.s, such as the common difference between consecutive terms and the formula for the sum of an A.P.

i. Determining the Value of m in the Arithmetic Progression

To find the value of m, we'll leverage the defining characteristic of an arithmetic progression: the difference between consecutive terms is constant. This constant difference is known as the common difference. Given the first three terms (m+1), (4m-2), and (6m-3), we can set up equations based on this property.

The common difference, d, can be expressed in two ways:

• d = (4m - 2) - (m + 1)
• d = (6m - 3) - (4m - 2)

Since the common difference is the same throughout the A.P., we can equate these two expressions:

(4m - 2) - (m + 1) = (6m - 3) - (4m - 2)

Now, let's simplify and solve for m. First, expand the expressions:

4m - 2 - m - 1 = 6m - 3 - 4m + 2

Combine like terms on both sides of the equation:

3m - 3 = 2m - 1

Next, isolate m by subtracting 2m from both sides:

3m - 2m - 3 = 2m - 2m - 1

m - 3 = -1

Finally, add 3 to both sides to solve for m:

m - 3 + 3 = -1 + 3

m = 2

Therefore, the value of m is 2. Now that we've found m, we can determine the actual terms of the A.P. by substituting m = 2 into the given expressions:

• First term (a): m + 1 = 2 + 1 = 3
• Second term: 4m - 2 = 4(2) - 2 = 8 - 2 = 6
• Third term: 6m - 3 = 6(2) - 3 = 12 - 3 = 9

So, the arithmetic progression begins with 3, 6, 9, and so on. We also know that the last term is 18. To find the sum of the terms, we first need to determine the number of terms in this A.P. This requires using the formula for the nth term of an A.P.

ii. Calculating the Sum of the Terms in the Arithmetic Progression

Having found the value of m to be 2, we've established the arithmetic progression as 3, 6, 9, ..., 18. To calculate the sum of the terms in this A.P., we first need to determine the number of terms present. We'll utilize the formula for the nth term of an A.P., which is given by:

an = a + (n - 1)d

where:

• an is the nth term (in our case, the last term, 18)
• a is the first term (3)
• n is the number of terms (what we want to find)
• d is the common difference (6 - 3 = 3)

Substituting the known values into the formula, we get:

18 = 3 + (n - 1)3

Now, we solve for n:

18 = 3 + 3n - 3

18 = 3n

n = 18 / 3

n = 6

Therefore, there are 6 terms in this arithmetic progression. Now that we know the number of terms, the first term (a = 3), and the last term (an = 18), we can use the formula for the sum of an arithmetic progression:

Sn = (n/2)(a + an)

where:

• Sn is the sum of the first n terms
• n is the number of terms (6)
• a is the first term (3)
• an is the last term (18)

Substituting the values, we get:

S6 = (6/2)(3 + 18)

S6 = 3(21)

S6 = 63

Therefore, the sum of the terms in the arithmetic progression is 63. In summary, we first determined the value of m by using the property of a constant common difference in an A.P. This allowed us to identify the terms of the progression. Then, we used the formula for the nth term to find the number of terms in the A.P. Finally, we applied the formula for the sum of an A.P. to calculate the sum of all the terms.

Beyond this specific problem, understanding arithmetic progressions is crucial in various mathematical and real-world scenarios. Let's explore some additional concepts and problem types related to A.P.s.

Finding a Specific Term in an A.P.

As we saw earlier, the formula an = a + (n - 1)d is essential for finding any specific term in an A.P. if you know the first term, the common difference, and the position of the term you're looking for. For instance, you might be asked to find the 20th term of an A.P. where the first term is 5 and the common difference is 2. In this case, a = 5, d = 2, and n = 20. Plugging these values into the formula gives you a20 = 5 + (20 - 1)2 = 5 + 38 = 43. Therefore, the 20th term is 43.

Inserting Arithmetic Means

Another common problem type involves inserting a certain number of arithmetic means between two given numbers. Arithmetic means are terms that, when inserted between two numbers, form an A.P. For example, if you need to insert three arithmetic means between 2 and 14, you're essentially creating an A.P. with five terms: 2, _, _, _, 14. To solve this, you first find the common difference. The first term (a) is 2, the last term (a5) is 14, and the number of terms (n) is 5. Using the formula an = a + (n - 1)d, you can solve for d: 14 = 2 + (5 - 1)d => 12 = 4d => d = 3. Now that you have the common difference, you can find the arithmetic means by adding the common difference successively: 2 + 3 = 5, 5 + 3 = 8, 8 + 3 = 11. So, the three arithmetic means between 2 and 14 are 5, 8, and 11.

Applications of Arithmetic Progressions

Arithmetic progressions aren't just abstract mathematical concepts; they have practical applications in various fields. For example, consider a simple interest problem where you deposit a fixed amount of money each month into an account. If the interest is calculated linearly, the amounts in your account each month will form an A.P. Similarly, the depreciation of an asset over time, if it depreciates by a fixed amount each year, can be modeled using an A.P.

Solving More Complex A.P. Problems

Some A.P. problems involve more intricate scenarios. You might be given the sum of the first n terms and the sum of the first m terms and asked to find the A.P. or a specific term. These problems often require setting up a system of equations and solving them simultaneously. A thorough understanding of the formulas and properties of A.P.s, combined with algebraic manipulation skills, is key to tackling these challenges.

Conclusion

Understanding arithmetic progressions is fundamental to grasping more advanced mathematical concepts. By mastering the formulas, properties, and problem-solving techniques associated with A.P.s, you'll be well-equipped to tackle a wide range of mathematical problems and real-world applications. The problem we initially addressed, finding the value of m and the sum of an A.P., exemplifies the core principles involved. By practicing various types of A.P. problems, you can strengthen your understanding and build your mathematical prowess.