The fifth term of a geometric progression is 16 and the tenth term is 54 .find the first term and common ratio and find the fifteenth term
there are 5 terms between 16 and 54, so r^5 = 54/16 = 27/8
This would have been a lot easier if it had been the 8th term, instead of the 10th, right? Then we'd have had r = 3/2
But now we have r = (3/2)^(3/5)
so a = 16 * r^-4 = 16 * (2/3)^(12/5)
a_15 = a_10 * r^5 = 54 * 27/8 = 729/4
To find the first term (a) and the common ratio (r) of a geometric progression, we'll use the formulas:
\[a_n = a \cdot r^{(n-1)}\]
\[a_n = a \cdot r^{(n-1)}\]
Given that the fifth term (a5) is 16 and the tenth term (a10) is 54, we can set up two equations using the formulas:
Equation 1:
\[a_5 = a \cdot r^{(5-1)} = 16\]
Equation 2:
\[a_{10} = a \cdot r^{(10-1)} = 54\]
Simplifying Equation 1:
\[16 = a \cdot r^4\]
\[a = \frac{16}{r^4} \hspace{30pt} (1)\]
Simplifying Equation 2:
\[54 = a \cdot r^9\]
\[a = \frac{54}{r^9} \hspace{30pt} (2)\]
Setting both equations equal to each other:
\[\frac{16}{r^4} = \frac{54}{r^9}\]
Cross-multiplying:
\[16 \cdot r^9 = 54 \cdot r^4\]
Rearranging the equation:
\[16 \cdot r^9 - 54 \cdot r^4 = 0\]
Now, let's solve this equation to find the value of r.
To find the first term and common ratio of a geometric progression, we can use the given information to set up a system of equations.
Let's denote the first term as 'a' and the common ratio as 'r'.
Given that the fifth term is 16, we can use the formula for the nth term of a geometric progression:
a * r^(n-1)
Substituting n = 5, we have:
a * r^(5-1) = 16
a * r^4 = 16 ---(equation 1)
Similarly, given that the tenth term is 54:
a * r^(10-1) = 54
a * r^9 = 54 ---(equation 2)
Now, we can solve this system of equations simultaneously.
Divide equation 2 by equation 1:
(a * r^9) / (a * r^4) = 54 / 16
r^5 = 27/8
Taking the fifth root of both sides to isolate 'r', we get:
r = (27/8)^(1/5)
Now, substitute the value of 'r' back into equation 1 to find 'a':
a * r^4 = 16
a * [(27/8)^(1/5)]^4 = 16
a * [(27^4)/(8^4)]^(1/5) = 16
a * (19683/4096)^(1/5) = 16
Taking the fifth root of both sides to isolate 'a', we have:
a = 16 * [(19683/4096)^(1/5)]
Now, you can use a calculator to find the approximate values of 'a' and 'r'.
Once you have determined the values of 'a' and 'r', you can find the fifteenth term by substituting n = 15 into the formula for the nth term:
a * r^(n-1)
Substituting the values, you can find the fifteenth term.