The third term of a GP is 9 and the fifth term is 16 find the fourth term, first term and the common ratio
To find the fourth term, first term, and common ratio of a geometric progression (GP), we can use the formulas given by:
\(a_n = a_1 \cdot r^{(n-1)}\)
where \(a_n\) is the \(n\)th term, \(a_1\) is the first term, and \(r\) is the common ratio.
Given that the third term (\(a_3\)) is 9 and the fifth term (\(a_5\)) is 16, we can use these values to find the fourth term, first term, and common ratio.
Step 1: Find the common ratio (r)
We can find the common ratio by dividing the fifth term by the third term:
\(r = \frac{{a_5}}{{a_3}} = \frac{{16}}{{9}}\)
Step 2: Find the first term (a1)
To find the first term, we can substitute the values of the third term, common ratio, and \(n\) into the \(a_n\) formula:
\(a_n = a_1 \cdot r^{(n-1)}\)
Substituting \(a_3 = 9\) and \(r = \frac{{16}}{{9}}\), we have:
\(9 = a_1 \cdot \left(\frac{{16}}{{9}}\right)^2\)
Simplifying this equation:
\(\frac{{9}}{{a_1}} = \left(\frac{{16}}{{9}}\right)^2\)
\(\frac{{9}}{{a_1}} = \frac{{256}}{{81}}\)
\(a_1 = \frac{{81}}{{256}} \cdot 9 = \frac{{9}}{{32}}\)
Therefore, the first term (a1) is \(a_1 = \frac{{9}}{{32}}\).
Step 3: Find the fourth term (a4)
To find the fourth term, we can substitute the values of the first term and common ratio into the \(a_n\) formula:
\(a_n = a_1 \cdot r^{(n-1)}\)
Substituting \(a_1 = \frac{{9}}{{32}}\) and \(r = \frac{{16}}{{9}}\), we have:
\(a_4 = \frac{{9}}{{32}} \cdot \left(\frac{{16}}{{9}}\right)^3\)
Simplifying this expression:
\(a_4 = \frac{{9}}{{32}} \cdot \frac{{4096}}{{729}} = \frac{{12}}{{1}} = 12\)
Therefore, the fourth term (a4) is 12.
To find the common ratio (r) of a geometric progression (GP), we can use the formula:
r = (n-th term) / (previous term)
Given that the third term is 9, we can use the formula to find the common ratio when n = 3:
r = 9 / (previous term)
To find the previous term, we can use the formula:
(previous term) = (previous n-th term) * r
Given that the fifth term is 16, we can use this formula twice to find the previous term:
(previous term) = 16 / r
(previous previous term) = (16 / r) / r
To find the fourth term, we again use the formula:
(fourth term) = (third term) * r
Now let's substitute the given values and solve the equations step by step:
1. Find the common ratio (r) using the third term:
r = 9 / (previous term)
2. Find the previous term using the fifth term:
(previous term) = 16 / r
3. Find the previous previous term:
(previous previous term) = (previous term) / r
4. Find the fourth term:
(fourth term) = (third term) * r
By following these steps, we can find the values of the fourth term, first term, and common ratio for the given geometric progression.
given:
ar^2 = 9
ar^4 = 16
divide the second by the first:
r^2 = 16/9
r = ± 4/3 <---- common ratio
ar^2 = 9
(16/9)a = 9
a = 81/16 <---- first term
check: if r = +4/3
terms would be 81/16, 27/4, 9, 12, 16, ..
if r + -4/3
terms would be 81/16, -27/4, 9, -12. 16
all looks good