in a GP the sum of first and fifth term is 34 and the product of second ,third and fifth term is 512.find the numbers.
a + ar^4 = 34
ar * ar^2 * ar^4 = 512
a(1+r^4) = 34
a^3 r^7 = 512
Looks kind of messy, but since the only factors of 34 are 2 and 17, it appears that
a=2 and r=2
I think the second condition should have used the 4th term, rather than the 5th. If so, then since 2^9=512, that fits with what we have so far.
To find the numbers in the geometric progression (GP), we need to use the given information in two different conditions: the sum of the first and fifth terms, and the product of the second, third, and fifth terms.
Let's denote the first term as 'a' and the common ratio as 'r'.
First, let's use the condition that the sum of the first and fifth terms is 34:
The first term: a
The fifth term: ar^4
Therefore, according to the condition, we have the equation:
a + ar^4 = 34 (equation 1)
Next, let's use the condition that the product of the second, third, and fifth terms is 512:
The second term: ar
The third term: ar^2
The fifth term: ar^4
Now, we can write another equation using this condition:
(ar) * (ar^2) * (ar^4) = 512
a^4 * r^7 = 512 (equation 2)
To solve these two equations simultaneously, we can divide equation 2 by equation 1 to eliminate 'a':
(a^4 * r^7) / (a + ar^4) = 512 / 34
a^3 * r^6 = 512 / 34
a^3 * r^6 = 16
At this point, we can try different values for 'a' and 'r' that satisfy the equation a^3 * r^6 = 16. One such combination is a = 2 and r = 2.
Plugging these values back into equation 1, we have:
2 + 2 * 2^4 = 34
2 + 2 * 16 = 34
2 + 32 = 34
34 = 34
So, the values of the numbers in the geometric progression are:
First term (a) = 2
Second term (ar) = 2 * 2 = 4
Third term (ar^2) = 2 * 2^2 = 8
Fourth term (ar^3) = 2 * 2^3 = 16
Fifth term (ar^4) = 2 * 2^4 = 32
Therefore, the numbers in the geometric progression are 2, 4, 8, 16, and 32.