The 6th term of a gp is 2000.find the first term if its common ratio is 10
In GP n-th term is:
an = a • r ⁿ⁻¹
where
a = a1 = the first term
r = common ratio
In this case:
n = 6
a6 = 2,000
r = 10
a6 = a • r ⁵
2,000 = a • 10 ⁵
2,000 = a • 100,000
100,000 a = 2,000
a = 2,000 / 100,000
a = 0.02
a1 = a = 0.02
To find the first term of a geometric progression (GP), we need to know the common ratio and either the term number or a specific term value.
In this case, we are given the common ratio, which is 10, and the value of the 6th term, which is 2000. We can use this information to find the first term.
The formula to find the nth term of a GP is: a_n = a * r^(n-1), where a_n is the nth term, a is the first term, r is the common ratio, and n is the term number.
Since we know the 6th term (a_6) is 2000 and the common ratio (r) is 10, we can substitute these values into the formula and calculate the first term (a).
a_6 = a * r^(n-1)
2000 = a * 10^(6-1)
2000 = a * 10^5
Now, we can solve for a by dividing both sides of the equation by 10^5:
2000 / (10^5) = a * 10^5 / (10^5)
0.02 = a
Therefore, the first term (a) is 0.02.
To find the first term of the geometric progression (GP), we can use the general formula for the nth term of a GP:
an = a * r^(n-1)
Given that the 6th term (an) is 2000 and the common ratio (r) is 10, we can substitute these values into the formula:
2000 = a * 10^(6-1)
Simplifying the equation:
2000 = a * 10^5
To find the value of a (the first term), we need to isolate it on one side of the equation. Divide both sides by 10^5:
2000 / 10^5 = a
Using a calculator to evaluate 2000 / 10^5, we get:
a ≈ 0.02
Therefore, the first term of the geometric progression is approximately 0.02.