The 8th term of a g.p is 640 if the first term is 5.find the common ration and the 10th term. o
r^7 = 640/5 = 128
r = 2
a_10 = ar^9 = 640*2^2 = ____
To find the common ratio of a geometric progression (g.p.), we can use the formula:
Tn = a * r^(n-1)
where Tn is the nth term, a is the first term, r is the common ratio, and n is the number of terms.
We are given the values T8 = 640 and a = 5. We want to find the common ratio r and the 10th term T10.
Step 1: Find the common ratio (r)
Let's substitute the given values into the formula:
640 = 5 * r^(8-1)
Simplify the equation:
640 = 5 * r^7
Divide both sides by 5:
128 = r^7
Take the 7th root of both sides:
∛128 = ∛(r^7)
Simplify:
2 = r
So, the common ratio (r) is 2.
Step 2: Find the 10th term (T10)
Now, we can substitute the values of a and r into the formula to find the 10th term:
T10 = 5 * 2^(10-1)
Simplify the exponent:
T10 = 5 * 2^9
Evaluate 2^9:
T10 = 5 * 512
Multiply:
T10 = 2560
The 10th term of the geometric progression is 2560.
To find the common ratio of a geometric progression (g.p.), we can use the formula:
nth term = a * r^(n-1)
where:
- nth term is the term we want to find
- a is the first term of the g.p.
- r is the common ratio of the g.p.
- n is the position of the term we want to find
In this case, we are given that the 8th term (n = 8) is 640 and the first term (a) is 5. We can plug these values into the formula to solve for the common ratio (r):
640 = 5 * r^(8-1)
Simplifying the equation, we get:
640 = 5 * r^7
To isolate r, we divide both sides of the equation by 5:
640/5 = r^7
128 = r^7
Now, to find the 10th term of the g.p., we can plug the values of the first term (a = 5), the common ratio (r), and the position of the term (n = 10) into the same formula:
10th term = 5 * r^(10-1)
Simplifying the equation, we get:
10th term = 5 * r^9
To calculate the common ratio (r) and the 10th term, we need to solve the equation 128 = r^7 for r and substitute the value of r into the equation for the 10th term.