The second term of a g.p. Is 27 and 7th term is 1/9. find the first trem 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:
nth term of a g.p.: a^n = a * r^(n-1)
n = term number
a = first term
r = common ratio
Given that the second term (n = 2) of the g.p. is 27, we can substitute these values into the formula:
27 = a * r^(2-1)
Simplifying this equation, we have:
27 = a * r
Similarly, the seventh term (n = 7) is 1/9:
1/9 = a * r^(7-1)
Simplifying this equation, we have:
1/9 = a * r^6
Now, we have a system of two equations with two variables:
Equation 1: 27 = a * r
Equation 2: 1/9 = a * r^6
To solve this system, we can divide Equation 2 by Equation 1:
(1/9) / 27 = (a * r^6) / (a * r)
1/243 = r^5
Taking the fifth root of both sides, we get:
r = (1/243)^(1/5)
Now, substitute this value of r back into Equation 1 to solve for a:
27 = a * (1/243)^(1/5)
To simplify further, we can convert 27 to a power of 3:
3^3 = a * (1/243)^(1/5)
Now, simplify the right side:
3^3 = a * (1/3)^(-1)
Apply the rule of negative exponents:
3^3 = a * 3
Simplify the left side:
27 = 3a
Divide both sides by 3:
a = 9
So, the first term of the g.p. is 9, and the common ratio (r) is approximately 0.3
To find the first term and the common ratio of a geometric progression (g.p.), we can use the formulas:
Formula for finding the nth term of a g.p.:
an = a1 * r^(n-1)
Formula for finding the common ratio (r) of a g.p.:
r = (an/an-1)
Given information:
Second term (a2) = 27
Seventh term (a7) = 1/9
Step 1: Find the common ratio (r):
Using the formula for finding the common ratio of a g.p.:
r = a2 / a1
Substituting the value of a2 = 27, we have:
r = 27 / a1
Step 2: Find the value of a7:
Using the formula for finding the nth term of a g.p.:
a7 = a1 * r^(7-1)
a7 = a1 * r^6
Substituting the value of a7 = 1/9, we have:
1/9 = a1 * r^6
Step 3: Combine the two equations to solve for a1:
From Step 1, we have: r = 27 / a1
Plugging this value into the equation from Step 3:
1/9 = a1 * (27 / a1)^6
Simplifying the equation:
1/9 = a1 * (27^6 / a1^6)
1/9 = 27^6 / a1^5
Multiply both sides of the equation by 9a1^5:
a1^5 = 9 * 27^6
Taking the fifth root of both sides:
a1 = ∛(9 * 27^6)
Calculating this value:
a1 ≈ 9 * 27^(6/5)
Hence, the first term (a1) is approximately equal to 9 * 27^(6/5) and the common ratio (r) is 27 / a1.