the 6th term of Gp is 2000.find its first term if its commom ratio is 10.
Use your definitions ..... Term(n) = a r^(n-1)
a(10^5) = 2000
solve for a
Term(n)=a r^(n-1) a(10^5)=2000
To find the first term of a geometric progression (GP) when the 6th term and common ratio are given, we can use the formula:
\[a_n = a_1 \cdot r^{(n-1)}\]
Where:
- \(a_n\) is the nth term of the GP,
- \(a_1\) is the first term of the GP,
- \(r\) is the common ratio of the GP,
- \(n\) is the position of the term in the GP.
In this case, the 6th term is given as 2000, and the common ratio is 10.
We can substitute these values into the formula and solve for \(a_1\):
\[2000 = a_1 \cdot 10^{(6-1)}\]
Simplifying the exponent:
\[2000 = a_1 \cdot 10^5\]
Dividing both sides of the equation by \(10^5\):
\[\frac{2000}{10^5} = a_1\]
Simplifying the fraction:
\[0.02 = a_1\]
Therefore, the first term of the geometric progression is 0.02.
To find the first term of a geometric progression (GP) when the common ratio and a specific term are given, you can use the formula:
n-th term = a * r^(n-1)
where:
- a represents the first term of the GP,
- r represents the common ratio of the GP,
- n represents the specific term of the GP that you want to find.
In this case, you are given that the 6th term of the GP is 2000, and the common ratio is 10. Let's substitute these values into the formula and solve for 'a':
2000 = a * 10^(6-1)
Simplifying the equation:
2000 = a * 10^5
Now divide both sides of the equation by 10^5:
2000 / (10^5) = a
Calculating this value:
0.02 = a
Therefore, the first term of the GP is 0.02.