Write a recursive formula and an explicit formula for each arithmetic sequence. 4. 9, 15, 21, 27, … 5. 1.5, 2.25, 3, 3.75, … 6. 7, 0, −7, −14,
#4. d = a_2 - a_1 = 15-9 = 6
a_1 = 9
a_n = a_n-1 + 6
a_n = 9 + 6(n-1) = 3+6n
Do the others in like wise
To find the recursive and explicit formulas for each arithmetic sequence, we need to determine the common difference between consecutive terms.
4. Arithmetic sequence: 9, 15, 21, 27, ...
Common difference (d) = 15 - 9 = 6
The recursive formula for this arithmetic sequence is:
a₁ = 9
aₙ = aₙ₋₁ + d
The explicit formula for this arithmetic sequence is:
aₙ = a₁ + (n-1)d
5. Arithmetic sequence: 1.5, 2.25, 3, 3.75, ...
Common difference (d) = 2.25 - 1.5 = 0.75
The recursive formula for this arithmetic sequence is:
a₁ = 1.5
aₙ = aₙ₋₁ + d
The explicit formula for this arithmetic sequence is:
aₙ = a₁ + (n-1)d
6. Arithmetic sequence: 7, 0, −7, −14, ...
Common difference (d) = 0 - 7 = -7
The recursive formula for this arithmetic sequence is:
a₁ = 7
aₙ = aₙ₋₁ + (-7)
The explicit formula for this arithmetic sequence is:
aₙ = a₁ + (n-1)(-7)
To find the recursive and explicit formulas for each arithmetic sequence, we need to examine the pattern of the sequence.
4. 9, 15, 21, 27, ...
To determine the recursive formula, we observe that each term is obtained by adding 6 to the previous term. Therefore, we can write:
a(n) = a(n-1) + 6
where a(n) represents the nth term and a(n-1) represents the previous term.
To find the explicit formula, we need to find the common difference, which is the amount added or subtracted to each term to obtain the next term. In this sequence, the common difference is 6.
Using this information, we can write the explicit formula:
a(n) = a(1) + (n-1)d
where a(1) is the first term, n is the position of the term, and d is the common difference. Substituting the values, we have:
a(n) = 9 + (n-1)6
5. 1.5, 2.25, 3, 3.75, ...
Observing this sequence, we can see that each term is obtained by multiplying the previous term by 1.5. Therefore, we can write the recursive formula as:
a(n) = a(n-1) * 1.5
To find the explicit formula, we examine the common ratio, which is the factor by which each term is multiplied to obtain the next term. In this sequence, the common ratio is 1.5.
Using this information, we can write the explicit formula as:
a(n) = a(1) * r^(n-1)
where a(1) is the first term, r is the common ratio, and n represents the position of the term. Substituting the values, we have:
a(n) = 1.5 * 1.5^(n-1)
6. 7, 0, -7, -14, ...
Examining this sequence, we observe that each term is obtained by subtracting 7 from the previous term. Therefore, we can write the recursive formula as:
a(n) = a(n-1) - 7
To find the explicit formula, we identify the common difference, which is the amount subtracted or added to each term to obtain the next term. In this sequence, the common difference is -7.
Using this information, we can write the explicit formula as:
a(n) = a(1) + (n-1)d
where a(1) is the first term, n represents the position of the term, and d is the common difference. Substituting the values, we have:
a(n) = 7 + (n-1)(-7)