Summing Terms in an Arithmetic Sequence: Finding the 26th and 54th Terms

Arithmetic sequences are fundamental in mathematics, and understanding how to find specific terms and the sum of terms is essential for various applications. This article explains how to calculate the sum of the terms between the 26th and 54th terms of an arithmetic sequence given specific terms.

Given Information:

We are given the 5th term and the 14th term of an arithmetic sequence:

Step 1: Determine the Common Difference

To find the common difference (d), we can use the formula:

a#8211;1 a5 (14 #8722; 5)d

Substituting the given values:

-1 -64 9d

By solving for d:

9d 63

d 7

Step 2: Determine the First Term (a)

Using the formula for the nth term of an arithmetic sequence:

an a (n-1)d

For the 5th term:

-64 a 4d

Substituting the common difference (d 7):

-64 a - 28

Solving for a:

a -92

Step 3: Calculate the 26th and 54th Terms

Using the same formula for the nth term:

a26 a 25d

a26 -92 25 * 7

a26 83

And for the 54th term:

a54 a 53d

a54 -92 53 * 7

a54 279

Step 4: Calculate the Sum of Terms from the 26th to the 54th Term

The sum of terms in an arithmetic sequence is given by:

Sn n/2 * (a1 an)

Here, the number of terms (n) is 54 - 26 1 29:

S 29/2 * (83 279)

S 29/2 * 362

S 5249

Conclusion:

By following these steps, we have determined that the sum of the terms between the 26th and 54th terms in the given arithmetic sequence is 5249.