In a G.P the product of the 2nd and 4th term is double the 5th term and the sum of the first 4th term is 80.find the first term
Let the first term of the G.P be 'a' and the common ratio be 'r'.
The second term is ar, the 4th term is ar^3, and the 5th term is ar^4.
Given, the product of the 2nd and 4th term is double the 5th term:
(ar)(ar^3) = 2(ar^4)
ar^4 = 2ar^4
r = 2
Thus, the G.P becomes: a, 2a, 4a, 8a, 16a, ...
Given, the sum of the first 4 terms is 80:
a + 2a + 4a + 8a = 80
15a = 80
a = 80/15
a = 16/3
Therefore, the first term of the G.P is 16/3.
(ar)(ar^3) = 2(ar^4)
a^2 r^4 = 2ar^4
a^2 = 2a
a = 2
so it does not matter what r is
we have answered the question, but let's go on.
now we know that
S4 = 2(r^4-1)/(r-1) = 80
(r^4-1)/(r-1) = 40
r = 3
so the sequence is 2,6,18,54,162,...
check: 6*54 = 2*162 ok
Apologies for the mistake in my previous response. Thank you for correcting it.
Starting with the original equation:
(ar)(ar^3) = 2(ar^4)
a^2 r^4 = 2ar^4
a^2 = 2a
a = 2
So, the first term of the G.P is indeed 2. As you correctly explained, the common ratio 'r' can be any value, as long as the first term is 2.
Moving on to finding the common ratio 'r':
The sum of the first 4 terms is 80:
S4 = 2(r^4-1)/(r-1) = 80
(r^4-1)/(r-1) = 40
By trial and error, you found that 'r' equals 3, which leads to the sequence 2, 6, 18, 54, 162,...
Checking the condition given in the question, we see that 6*54 = 2*162, satisfying the product relationship. Once again, thank you for pointing out the error, and I appreciate your clarification.