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QuestionFind the explicit formulas for the sequences with:
9. a1=−1a_1 = -1, d=−10d = -10
10. a1=−3a_1 = -3, d=−9d = -9

Studdy Solution

STEP 1

Assumptions1. We are dealing with arithmetic sequences. . The first term of the sequence in problem9 is -1, denoted as a1=−1a1 = -1.
3. The common difference of the sequence in problem9 is -10, denoted as d=−10d = -10.
4. The first term of the sequence in problem10 is -3, denoted as a1=−3a1 = -3.
5. The common difference of the sequence in problem10 is -9, denoted as d=−9d = -9.

STEP 2

The explicit formula for an arithmetic sequence is given byan=a1+(n−1)⋅da_n = a1 + (n-1) \cdot dwhere ana_n is the nth term, a1a1 is the first term, dd is the common difference, and nn is the term number.

STEP 3

For problem9, plug in the given values for the first term and the common difference into the formula.
an=−1+(n−1)⋅−10a_n = -1 + (n-1) \cdot -10

STEP 4

implify the formula for problem9.
an=−1−10(n−1)a_n = -1 -10(n-1)

STEP 5

Further simplify the formula for problem9.
an=−1−10n+10a_n = -1 -10n +10

STEP 6

Rearrange the terms to get the final explicit formula for problem9.
an=9−10na_n =9 -10n

STEP 7

For problem10, plug in the given values for the first term and the common difference into the formula.
an=−3+(n−1)⋅−9a_n = -3 + (n-1) \cdot -9

STEP 8

implify the formula for problem10.
an=−3−(n−1)a_n = -3 -(n-1)

STEP 9

Further simplify the formula for problem.
an=−3−9n+9a_n = -3 -9n +9

STEP 10

Rearrange the terms to get the final explicit formula for problem10.
an=6−9na_n =6 -9nSo, the explicit formulas for the given arithmetic sequences are an=9−10na_n =9 -10n and an=6−9na_n =6 -9n respectively.

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