Here are the solutions to the problems on your worksheet.
1. Write a recursive formula for the following sequences: 2, 5, 26, 677...
To find the pattern, look at how each number relates to the one before it:
* $2^2 + 1 = 5$
* $5^2 + 1 = 26$
* $26^2 + 1 = 677$
The rule is: square the previous term and add 1.
Recursive Formula:
$t_1 = 2$
$t_n = (t_{n-1})^2 + 1$
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2. Given the recursive formula, write the explicit formula for the sequence: $t_1 = 0, t_n = t_{n-1} - 2$
This is an arithmetic sequence because we are subtracting the same number (2) each time.
* First term ($a$): 0
* Common difference ($d$): -2
Explicit formula structure: $t_n = a + d(n - 1)$
Substitute the values: $t_n = 0 + (-2)(n - 1)$
Simplify: $t_n = -2(n - 1)$ or $t_n = -2n + 2$
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3. Write an explicit formula for the following sequences: 20, 24, 28, 32, 36...
First, find the common difference by subtracting the first term from the second: $24 - 20 = 4$.
* First term ($a$): 20
* Common difference ($d$): 4
Formula: $t_n = 20 + 4(n - 1)$
Distribute the 4: $t_n = 20 + 4n - 4$
Combine like terms:
$t_n = 4n + 16$
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4. Write an explicit formula for the following sequences: 3, 7, 11, 15, 19...
Find the common difference: $7 - 3 = 4$.
* First term ($a$): 3
* Common difference ($d$): 4
Formula: $t_n = 3 + 4(n - 1)$
Distribute the 4: $t_n = 3 + 4n - 4$
Combine like terms:
$t_n = 4n - 1$
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5. Given the explicit formula, write the recursive formula for the sequence: $T_n = 2n - 1$
First, find the first term ($t_1$) by plugging in $n=1$:
$t_1 = 2(1) - 1 = 1$
Next, find the common difference. Since the formula is linear ($2n$), the difference is 2. You can check this by finding the second term ($t_2 = 3$) and seeing that $3 - 1 = 2$.
Recursive Formula:
$t_1 = 1$
$t_n = t_{n-1} + 2$
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6. Given the recursive formula, write the explicit formula for the sequence: $t_1 = 0, t_n = t_{n-1} - 5$
This is an arithmetic sequence with subtraction.
* First term ($a$): 0
* Common difference ($d$): -5
Formula: $t_n = 0 + (-5)(n - 1)$
Simplify:
$t_n = -5(n - 1)$ or
$t_n = -5n + 5$
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7. Write a recursive formula for the following sequences: 2, 5, 26, 677...
(This is the same sequence as Problem 1).
Recursive Formula:
$t_1 = 2$
$t_n = (t_{n-1})^2 + 1$
Parent Tip: Review the logic above to help your child master the concept of recursive formula worksheet with answers.
