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Math  /  Algebra

Question00110.0001 \quad 10.0 points For an arithmetic sequence a1=13 and d=−12.a_{1}=13 \text { and } d=-\frac{1}{2} .
Find the term a23a_{23} of the sequence. 00210.0002 \quad 10.0 points Consider an arithmetic sequence where a1=10 and d=5. a_{1}=10 \text { and } d=5 \text {. }
Find the term a27a_{27} of the sequence.

Studdy Solution

STEP 1

1. We are dealing with arithmetic sequences.
2. The formula for the n n -th term of an arithmetic sequence is an=a1+(n−1)⋅d a_n = a_1 + (n-1) \cdot d , where a1 a_1 is the first term and d d is the common difference.

STEP 2

1. Calculate a23 a_{23} for the first sequence.
2. Calculate a27 a_{27} for the second sequence.

STEP 3

For the first sequence, use the formula for the n n -th term of an arithmetic sequence:
Given: a1=13 a_1 = 13 d=−12 d = -\frac{1}{2} n=23 n = 23
Substitute these values into the formula: a23=a1+(n−1)⋅d a_{23} = a_1 + (n-1) \cdot d a23=13+(23−1)⋅(−12) a_{23} = 13 + (23-1) \cdot \left(-\frac{1}{2}\right)

STEP 4

Calculate the expression:
a23=13+22⋅(−12) a_{23} = 13 + 22 \cdot \left(-\frac{1}{2}\right) a23=13−11 a_{23} = 13 - 11 a23=2 a_{23} = 2

STEP 5

For the second sequence, use the formula for the n n -th term of an arithmetic sequence:
Given: a1=10 a_1 = 10 d=5 d = 5 n=27 n = 27
Substitute these values into the formula: a27=a1+(n−1)⋅d a_{27} = a_1 + (n-1) \cdot d a27=10+(27−1)⋅5 a_{27} = 10 + (27-1) \cdot 5

STEP 6

Calculate the expression:
a27=10+26⋅5 a_{27} = 10 + 26 \cdot 5 a27=10+130 a_{27} = 10 + 130 a27=140 a_{27} = 140
The term a23 a_{23} of the first sequence is 2 \boxed{2} and the term a27 a_{27} of the second sequence is 140 \boxed{140} .

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