Math

Question Find a recursive formula for the number of seats in each row of a movie theater, where the explicit formula is an=6+6na_{n}=6+6 n and nn represents the row number.

Studdy Solution

STEP 1

Assumptions
1. The explicit formula for the number of seats in each row is given by an=6+6na_{n}=6+6n.
2. The variable nn represents the row number.
3. We need to find the recursive formula, which defines each term based on the previous term(s).

STEP 2

Understand the explicit formula. The explicit formula an=6+6na_{n}=6+6n shows that the number of seats increases by 6 for each subsequent row.

STEP 3

Determine the first term of the sequence (a1a_1) by plugging in n=1n=1 into the explicit formula.
a1=6+6(1)a_{1} = 6 + 6(1)

STEP 4

Calculate the first term.
a1=6+6=12a_{1} = 6 + 6 = 12

STEP 5

Determine the second term of the sequence (a2a_2) by plugging in n=2n=2 into the explicit formula.
a2=6+6(2)a_{2} = 6 + 6(2)

STEP 6

Calculate the second term.
a2=6+12=18a_{2} = 6 + 12 = 18

STEP 7

Notice the pattern that each term is 6 more than the previous term. This is the basis for the recursive formula.

STEP 8

Write the recursive formula starting with the first term. The first term is a1=12a_1 = 12.
a1=12a_{1} = 12

STEP 9

Express the nn-th term ana_n as 6 more than the previous term an1a_{n-1}.
an=an1+6a_{n} = a_{n-1} + 6

STEP 10

Combine the information from steps 8 and 9 to write the complete recursive formula.
The recursive formula is: a1=12a_{1} = 12 an=an1+6 for n>1a_{n} = a_{n-1} + 6 \text{ for } n > 1
This recursive formula shows that the first term is 12 and each subsequent term is 6 more than the previous term.

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