Question

The 5th term in a sequence is 25 and each term is 3 less than the previous term. Write an explicit rule and recursive rule to describe the sequence

Answer 1

The explicit rule for the sequence is a_n = 40 - 3n, and the recursive rule is a_1 = 37, a_n = a_(n-1) - 3 for n > 1.

Step-by-step explanation:

To write an explicit rule and a recursive rule for a sequence where the 5th term is 25 and each term is 3 less than the previous term, we start by identifying the common difference and the first term.

Step 1: Since each term decreases by 3, the common difference (d) is -3.

Step 2: To find the first term (a_1), we can work backwards from the 5th term (a_5 = 25):

• a_4 = a_5 + 3 = 25 + 3 = 28
• a_3 = a_4 + 3 = 28 + 3 = 31
• a_2 = a_3 + 3 = 31 + 3 = 34
• a_1 = a_2 + 3 = 34 + 3 = 37

The first term (a_1) is 37.

Explicit rule:

To write the explicit rule, we use the formula for the nth term of an arithmetic sequence:

a_n = a_1 + (n - 1)d

Substituting the values we have:

a_n = 37 + (n - 1)(-3)

a_n = 37 - 3n + 3

a_n = 40 - 3n

Recursive rule:

For the recursive rule, we state the initial term and the method of finding each term:

a_1 = 37

a_n = a_(n-1) - 3 (for n > 1)