Introduction to Arithmetic Sequences

An arithmetic sequence is a sequence of numbers such that the difference between any two successive members is constant. In other words, each term after the first is derived from the previous one by adding a fixed number, known as the common difference.

Understanding the Given Sequence -8n - 1

The given sequence is defined by the formula a_n -8n - 1. Here, each term in the sequence depends on n, the position of the term in the sequence. Specifically, -8 is the common difference, and -1 is the constant term subtracted from the product of -8 and n.

Calculating the First Five Terms

To find the first five terms of the sequence, we need to substitute the values of n from 1 to 5 into the formula a_n -8n - 1.

tFor n 1: t(a_1 -8 cdot 1 - 1 -8 - 1 -9) tFor n 2: t(a_2 -8 cdot 2 - 1 -16 - 1 -17) tFor n 3: t(a_3 -8 cdot 3 - 1 -24 - 1 -25) tFor n 4: t(a_4 -8 cdot 4 - 1 -32 - 1 -33) tFor n 5: t(a_5 -8 cdot 5 - 1 -40 - 1 -41)

Therefore, the first five terms of the sequence are: -9, -17, -25, -33, -41.

Verification Using Summation

If we need to verify the summation of the first five terms, we can calculate the sum as:

(S -9 - 17 - 25 - 33 - 41 -125)

This confirms the calculation of the first five terms.

Algorithm Explanation

The formula for the nth term of an arithmetic sequence can be written as:

(a_n A (n-1)d)

where (A) is the first term, (d) is the common difference, and n is the position of the term.

In the sequence -8n - 1, we have:

tThe common difference ((d)) is -8. tThe first term ((A)) can be found by substituting n 1 into the formula: t(A -8 cdot 1 - 1 -9)

Thus, the first five terms using the formula are:

t(a_1 -9) t(a_2 -9 (-8) -17) t(a_3 -17 (-8) -25) t(a_4 -25 (-8) -33) t(a_5 -33 (-8) -41)

Matching these to the original calculation, we confirm that the terms are correct.

Conclusion

The first five terms of the sequence defined by a_n -8n - 1 are indeed -9, -17, -25, -33, -41. This sequence follows the arithmetic progression pattern with a common difference of -8.