How to Find the Tenth Term of an Arithmetic Progression Given Its Third and Seventh Terms

In this article, we will walk you through the process of finding the tenth term of an arithmetic progression (AP) when given the third and seventh terms. Understanding how to solve this type of problem is essential for anyone working with sequences and series, especially in mathematics.

Understanding the Arithmetic Progression Formula

In an arithmetic progression, the nth term can be expressed as:

an an-1 d

Where:

an is the nth term of the AP. a1 is the first term of the AP. d is the common difference between consecutive terms.

Given Information

For the problem at hand, we are given the following information:

The third term, a3 17 The seventh term, a7 37

Step-by-Step Solution

Step 1: Express the Third and Seventh Terms Using the General Formula

The general formula for the nth term of an AP is:

an a1 (n-1)d

Therefore, we can write:

a3 a1 2d 17 a7 a1 6d 37

Step 2: Solve the Simultaneous Equations

To find the common difference (d), we can subtract the first equation from the second:

a7 - a3 (a1 6d) - (a1 2d) 37 - 17

Simplifying, we get:

4d 20

Taking the common difference from both sides:

d 5

Step 3: Find the First Term

Substitute d 5 back into the equation for the third term:

a3 a1 2d a1 2(5) 17

Solving for a1:

a1 17 - 10 7

Step 4: Calculate the Tenth Term

The formula for the nth term is:

an a1 (n-1)d

For the tenth term, substituting n 10, a1 7, and d 5:

a10 7 9(5) 7 45 52

Conclusion

The tenth term of the given arithmetic progression is 52. This method of solving for unknown terms in an AP is widely applicable and can be useful in various mathematical and real-world contexts.

Re-desision Formulas and Simplified Calculations

The tenth term of the arithmetic progression can also be calculated using a simplified approach:

T3 17 frac{7 - 317 - 337}{7 - 3} and T7 17 frac{7 - 717 - 337}{7 - 3} so

T10 frac{7 - 1017 - 1037}{7 - 3} frac{208}{4} 52

This formula simplifies the process and can be particularly useful when dealing with larger terms or more complex arithmetic progressions.