Fourth term of a G.P. is 40 and the tenth term is 2560. find the seventh term.
Well, let's see. In a geometric progression (G.P.), each term is found by multiplying the previous term by a common ratio. So, to find the common ratio, we can divide the 10th term by the 4th term: 2560/40 = 64.
Now, since we know the common ratio is 64, we can use that to calculate the 7th term. Starting from the 4th term, let's multiply it by the common ratio three more times:
40 * 64 * 64 * 64 = 10485760.
So, the 7th term of the G.P. is 10,485,760. That's quite a large number! Just be careful not to fall off the numerical rollercoaster.
To find the seventh term of a geometric progression (G.P.), we need to identify the common ratio (r) first.
Given:
Fourth term (a₄) = 40
Tenth term (a₁₀) = 2560
We can use the formula for the nth term of a G.P.:
aₙ = a₁ * r^(n-1)
For the fourth term:
40 = a₁ * r^(4-1)
40 = a₁ * r^3
Similarly, for the tenth term:
2560 = a₁ * r^(10-1)
2560 = a₁ * r^9
Divide the equation of the tenth term by the fourth term to eliminate a₁:
2560 / 40 = (a₁ * r^9) / (a₁ * r^3)
64 = r^6
Now, take the 6th root of both sides to solve for r:
r = ∛(64)
r = 2
Substituting the value of r into the equation for the fourth term:
40 = a₁ * 2^3
40 = 8a₁
a₁ = 40 / 8
a₁ = 5
Now, we can calculate the seventh term using the value of a₁ and r:
a₇ = a₁ * r^(7-1)
a₇ = 5 * 2^6
a₇ = 5 * 64
a₇ = 320
Therefore, the seventh term of the given geometric progression is 320.
To find the seventh term of a geometric progression (G.P.), we need to find the common ratio and then use it to calculate the seventh term.
A geometric progression is a sequence of numbers where each term is multiplied by a constant factor to obtain the next term.
Let's assume that the first term of the G.P. is 'a', and the common ratio is 'r'.
Given information:
The fourth term of the G.P. is 40, so we can write this as:
a * r^3 = 40 ---(1)
The tenth term of the G.P. is 2560, so we can write this as:
a * r^9 = 2560 ---(2)
Now, let's divide equation (2) by equation (1) in order to eliminate the 'a' term:
(a * r^9) / (a * r^3) = 2560 / 40
r^6 = 64
Taking the sixth root of both sides, we get:
r = 2
Now, substitute the value of 'r' (i.e., 2) into equation (1) to find 'a':
a * 2^3 = 40
a * 8 = 40
a = 40 / 8
a = 5
We have found the first term 'a' to be 5 and the common ratio 'r' to be 2.
To find the seventh term, we can now use the formula for a term of a G.P.:
nth term = a * r^(n-1)
Substituting the values, we get:
seventh term = 5 * 2^(7-1)
seventh term = 5 * 2^6
seventh term = 5 * 64
seventh term = 320
Therefore, the seventh term of the G.P. is 320.