If the second term of a g.p. is 27 and 7th term is 1/9. Find the first term and the common ratio?
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To find the first term and the common ratio of a geometric progression (g.p.), we can use the following formulas:
The formula for the nth term of a g.p. is given by:
an = a1 * r^(n-1)
where:
an = nth term of the g.p.
a1 = first term of the g.p.
r = common ratio of the g.p.
n = term number in the g.p.
We are given that the second term (n = 2) is 27 and the seventh term (n = 7) is 1/9.
Step 1: Finding the common ratio (r)
Using the formula for the second term, we have:
27 = a1 * r^(2 - 1)
27 = a1 * r
Step 2: Finding the first term (a1)
Using the formula for the seventh term, we have:
1/9 = a1 * r^(7 - 1)
1/9 = a1 * r^6
Step 3: Solving the system of equations
We can now solve the system of equations formed by the equations found in steps 1 and 2.
Divide the equation from Step 2 by the equation from Step 1:
(1/9) / 27 = (a1 * r^6) / (a1 * r)
1/243 = r^5
Take the fifth root of both sides:
∛(1/243) = r
Simplifying:
1/3 = r
Now that we have the value of the common ratio (r), we can substitute it back into either Step 1 or Step 2 to find the first term (a1).
Using the equation from Step 1:
27 = a1 * (1/3)
27 = a1/3
Multiply both sides by 3 to isolate a1:
81 = a1
Therefore, the first term (a1) is 81 and the common ratio (r) is 1/3.
the second term of a g.p. is 27 ---> ar = 27
7th term is 1/9 ---> ar^6 = 1/9
divide them
(ar^6)/(ar) = (1/9) / 27
r^5 = 1/243 , and I know 3^5 = 243
continue