Recursive and Explicit Formulas Worksheet
Explore recursive and explicit formulas for sequences, including arithmetic and geometric progressions, with this Grade 11 math worksheet.
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Recursive and Explicit Formulas
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Read each question carefully and provide the requested formula or solution. Show all your work for full credit.
1. Given the recursive formula $a_1 = 3$ and $a_n = 2a_{n-1} + 1$ for $n > 1$, find the first five terms of the sequence.
2. Write a recursive formula for the sequence: 5, 8, 11, 14, ...
3. The explicit formula for an arithmetic sequence is $a_n = a_1 + (n-1)d$. If the first term is 7 and the common difference is 4, the 10th term is .
4. The explicit formula for a geometric sequence is $a_n = a_1 \cdot r^{n-1}$. If the first term is 2 and the common ratio is 3, the 5th term is .
5. Which of the following is the explicit formula for the recursive sequence $a_1 = 4$, $a_n = a_{n-1} + 5$?
$a_n = 4n + 5$
$a_n = 5n - 1$
$a_n = 5n + 4$
$a_n = 4 + 5(n-1)$
6. The explicit formula for a sequence is $a_n = 3 \cdot 2^{n-1}$. What is the recursive formula for this sequence?
$a_1 = 3, a_n = a_{n-1} + 2$
$a_1 = 3, a_n = 2a_{n-1}$
$a_1 = 2, a_n = 3a_{n-1}$
$a_1 = 2, a_n = a_{n-1} + 3$
7. A population of bacteria doubles every hour. If you start with 100 bacteria, write both a recursive and an explicit formula to model the population after 'n' hours. Then, find the population after 5 hours.
8. An arithmetic sequence has a common ratio.
True
False
9. A geometric sequence has a common difference.
True
False
Match each sequence with its corresponding explicit formula.
10. 2, 4, 6, 8, ...
a. $a_n = 3n + 1$
11. 4, 7, 10, 13, ...
b. $a_n = 2n$
12. 1, 3, 9, 27, ...
c. $a_n = 3^{n-1}$
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