if the first and 8th term of a g.p.is x^-4 and x^52 and the second term is x^t then t:
To find the value of 't', we need to understand the relationship between the terms of a geometric progression (g.p.). In a g.p., each term is obtained by multiplying the preceding term by a constant factor called the common ratio (r).
Let's consider the given g.p. with the first term (a) as x^-4 and the eighth term (b) as x^52.
The general formula for the nth term of a g.p. is:
a_n = a * r^(n-1)
In this case, we have a_1 = x^-4 and a_8 = x^52. Now, we can find the common ratio (r) by equating the two equations:
a_8 = a_1 * r^(8-1)
x^52 = (x^-4) * r^7
To simplify further, we can convert all the terms to a common base (let's choose x).
x^(52) = (x^(-4)) * (x^7)^r
x^(52) = x^(-4 + 7r)
By comparing the exponents, we get:
52 = -4 + 7r
Now, we can solve this equation to find the value of 'r':
7r = 52 + 4
7r = 56
r = 8
Now that we have the common ratio (r = 8), we can find the value of the second term (x^t).
Using the formula for the nth term of a g.p., we have:
a_2 = a_1 * r^(2-1)
x^t = (x^-4) * 8^(2-1)
x^t = (x^-4) * 8
To simplify, we can multiply the numbers:
x^t = 8x^-4 = (8/x^4)
Therefore, the value of 't' is -4.
seven steps and 56 powers of x
... so 8 powers per step
t = -4 + 8
given:
a= x^-4
ar^7 = x^52
use substitution:
(x^-4)r^7 = x^52
r^7 = x^59
r = x^(59/7)
2nd term = ar
= (x^-4)(x^59/7)
= x^(31/7)
matching it with x^t ---> t = 31/7