Math  /  Algebra

QuestionFor a given arithmetic sequence, the common difference, dd, is equal to 5 , and the 12th 12^{\text {th }} term, a12a_{12}, is equal to 40 . Find the value of the 88th 88^{\text {th }} term,a88a_{88}. a88=a_{88}=

Studdy Solution

STEP 1

1. We are dealing with an arithmetic sequence.
2. The common difference d d is 5.
3. The 12th term, a12 a_{12} , is 40.
4. We need to find the 88th term, a88 a_{88} .

STEP 2

1. Recall the formula for the n n -th term of an arithmetic sequence.
2. Use the given information to find the first term a1 a_1 .
3. Use the first term and the common difference to find the 88th term.

STEP 3

Recall the formula for the n n -th term of an arithmetic sequence:
an=a1+(n1)d a_n = a_1 + (n-1) \cdot d

STEP 4

Use the given information to find the first term a1 a_1 .
Given that a12=40 a_{12} = 40 , we can write:
a12=a1+(121)5 a_{12} = a_1 + (12-1) \cdot 5 40=a1+115 40 = a_1 + 11 \cdot 5 40=a1+55 40 = a_1 + 55
Subtract 55 from both sides to solve for a1 a_1 :
a1=4055 a_1 = 40 - 55 a1=15 a_1 = -15

STEP 5

Use the first term and the common difference to find the 88th term.
Using the formula for the n n -th term:
a88=a1+(881)5 a_{88} = a_1 + (88-1) \cdot 5 a88=15+875 a_{88} = -15 + 87 \cdot 5
Calculate 875 87 \cdot 5 :
875=435 87 \cdot 5 = 435
Now substitute back:
a88=15+435 a_{88} = -15 + 435 a88=420 a_{88} = 420
The value of the 88th term is:
420 \boxed{420}

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