In An Arithmetic Progression, The Sum Of The First 5 Terms Is 1550, And The Sum Of The Next 10 Terms Is 325. Find:(i) The First Term And The Common Difference.(ii) The Sum Of The First 13 Terms Of The Progression.

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Introduction

An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference. In this article, we will explore how to find the first term and the common difference of an arithmetic progression given the sum of the first few terms. We will also learn how to find the sum of the first 13 terms of the progression.

The Formula for the Sum of the First n Terms of an Arithmetic Progression

The formula for the sum of the first n terms of an arithmetic progression is given by:

S_n = n/2 * [2a + (n-1)d]

where S_n is the sum of the first n terms, a is the first term, n is the number of terms, and d is the common difference.

Given Information

We are given that the sum of the first 5 terms is 1550, and the sum of the next 10 terms is 325. We can use this information to set up two equations and solve for the first term and the common difference.

Equation 1: Sum of the First 5 Terms

Let's use the formula for the sum of the first n terms to write an equation for the sum of the first 5 terms:

S_5 = 5/2 * [2a + (5-1)d] = 1550

Simplifying the equation, we get:

2.5(2a + 4d) = 1550

Expanding and simplifying further, we get:

5a + 10d = 1550

Equation 2: Sum of the Next 10 Terms

The sum of the next 10 terms is equal to the sum of the first 15 terms minus the sum of the first 5 terms. We can use the formula for the sum of the first n terms to write an equation for the sum of the first 15 terms:

S_15 = 15/2 * [2a + (15-1)d] = S_5 + 325

Simplifying the equation, we get:

7.5(2a + 14d) = 1550 + 325

Expanding and simplifying further, we get:

15a + 105d = 1875

Solving the System of Equations

We now have a system of two equations with two unknowns (a and d). We can solve this system using substitution or elimination. Let's use the elimination method.

First, we can multiply Equation 1 by 3 to get:

15a + 30d = 4650

Now, we can subtract Equation 2 from this new equation to eliminate the variable a:

(15a + 30d) - (15a + 105d) = 4650 - 1875

Simplifying, we get:

-75d = 2775

Dividing both sides by -75, we get:

d = -37

Finding the First Term

Now that we have found the common difference, we can substitute this value into one of the original equations to find the first term. Let's use Equation 1:

5a + 10d = 1550

Substituting d = -37, we get:

5a + 10(-37) = 1550

Simplifying, we get:

5a - 370 = 1550

Adding 370 to both sides, we get:

5a = 1920

Dividing both sides by 5, we get:

a = 384

The First Term and the Common Difference

We have found that the first term (a) is 384 and the common difference (d) is -37.

The Sum of the First 13 Terms

Now that we have found the first term and the common difference, we can use the formula for the sum of the first n terms to find the sum of the first 13 terms:

S_13 = 13/2 * [2a + (13-1)d]

Substituting a = 384 and d = -37, we get:

S_13 = 6.5 * [2(384) + (12)(-37)]

Simplifying, we get:

S_13 = 6.5 * [768 - 444]

Simplifying further, we get:

S_13 = 6.5 * 324

Multiplying, we get:

S_13 = 2106

Conclusion

Q: What is an arithmetic progression?

A: An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference.

Q: How do I find the first term and the common difference of an arithmetic progression?

A: To find the first term and the common difference, you can use the formula for the sum of the first n terms of an arithmetic progression. You can set up a system of equations using the given information and solve for the unknowns.

Q: What is the formula for the sum of the first n terms of an arithmetic progression?

A: The formula for the sum of the first n terms of an arithmetic progression is given by:

S_n = n/2 * [2a + (n-1)d]

where S_n is the sum of the first n terms, a is the first term, n is the number of terms, and d is the common difference.

Q: How do I find the sum of the first n terms of an arithmetic progression?

A: To find the sum of the first n terms, you can use the formula:

S_n = n/2 * [2a + (n-1)d]

Substitute the values of a, n, and d into the formula and simplify to find the sum.

Q: What is the difference between an arithmetic progression and a geometric progression?

A: An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. A geometric progression is a sequence of numbers in which the ratio between any two consecutive terms is constant.

Q: How do I find the nth term of an arithmetic progression?

A: To find the nth term of an arithmetic progression, you can use the formula:

a_n = a + (n-1)d

where a_n is the nth term, a is the first term, n is the term number, and d is the common difference.

Q: What is the sum of an infinite arithmetic progression?

A: The sum of an infinite arithmetic progression is given by:

S = a / (1 - r)

where S is the sum, a is the first term, and r is the common ratio.

Q: How do I find the common ratio of an arithmetic progression?

A: To find the common ratio, you can use the formula:

r = d / a

where r is the common ratio, d is the common difference, and a is the first term.

Q: What are some real-world applications of arithmetic progressions?

A: Arithmetic progressions have many real-world applications, including:

  • Finance: calculating interest rates and investment returns
  • Music: calculating the frequency of notes in a musical scale
  • Science: calculating the velocity of an object in motion
  • Engineering: calculating the stress and strain on a material

Q: How do I use arithmetic progressions in real-world problems?

A: To use arithmetic progressions in real-world problems, you can:

  • Identify the sequence of numbers and determine the common difference
  • Use the formula for the sum of the first n terms to calculate the total
  • Use the formula for the nth term to find a specific term in the sequence
  • Use the formula for the sum of an infinite arithmetic progression to calculate the total for an infinite sequence.