solution to the fourth term of an exponential sequence is 108 and the common ratio is 3. Calculate the value of the first and eight terms of the sequence
since r=3, and ar^3 = 108, a=4
the 8th term is ar^7 = 108 * 3^4 = ____
To find the value of the first term and the eighth term of an exponential sequence, we need to use the formula:
\[a_n = a_1 \times r^{(n-1)},\]
where:
\(a_n\) is the \(n\)th term of the sequence,
\(a_1\) is the first term of the sequence,
\(r\) is the common ratio of the sequence, and
\(n\) is the position of the term in the sequence.
Given that the fourth term of the sequence is 108 and the common ratio is 3, we can substitute these values into the formula and solve for \(a_1\) and \(a_8\).
1. To find the first term, substitute \(n = 1\) and \(a_n = 108\) into the formula:
\[108 = a_1 \times 3^{(1-1)}.\]
Simplifying, we have:
\[a_1 = 108.\]
So, the value of the first term (\(a_1\)) is 108.
2. To find the eighth term, substitute \(n = 8\) and \(a_1 = 108\) into the formula:
\[a_8 = 108 \times 3^{(8-1)}.\]
Simplifying, we have:
\[a_8 = 108 \times 3^7.\]
Calculating the value of \(a_8\), we get:
\[a_8 = 108 \times 2187 = 236,196.\]
So, the value of the eighth term (\(a_8\)) is 236,196.
Therefore, the first term of the sequence is 108 and the eighth term is 236,196.
To find the value of the first term of the exponential sequence, we can use the formula:
aᵣ = a₁ * r^(r-1),
where aᵣ is the value of the rth term, a₁ is the first term, and r is the position of the term.
Given:
a₄ = 108, and
r = 4.
We can rearrange the formula and solve for a₁:
108 = a₁ * 3^(4-1)
108 = a₁ * 3^3
108 = a₁ * 27
a₁ = 108 / 27
a₁ = 4
So, the value of the first term (a₁) is 4.
To find the value of the eighth term, we can use the same formula:
aᵣ = a₁ * r^(r-1),
where r = 8.
a₈ = 4 * 3^(8-1)
a₈ = 4 * 3^7
a₈ = 4 * 2187
a₈ = 8748
So, the value of the eighth term (a₈) is 8748.