. If 2nd and 5th term of a G.P. is 24 and 81 then find
i. First and common ratio
ii. Find 10th term
iii. Find geometric series.
To find the first and common ratio in a geometric progression (G.P.), we can use the formulas provided in the question.
Given:
2nd term (a2) = 24
5th term (a5) = 81
i. First and common ratio:
We know that in a G.P., the nth term (an) can be expressed as:
an = a1 * r^(n-1)
Using this formula, we can find the first term (a1) and the common ratio (r) by setting up two equations from the given values:
Equation 1: a2 = a1 * r^(2-1)
24 = a1 * r
Equation 2: a5 = a1 * r^(5-1)
81 = a1 * r^4
To solve these equations, we will divide Equation 2 by Equation 1:
(81 / 24) = (a1 * r^4) / (a1 * r)
Simplifying:
(81 / 24) = r^3
Cube rooting both sides:
(81 / 24)^(1/3) = r
Calculating this:
r ≈ 1.5
Now, substituting the value of r back into Equation 1 to find a1:
24 = a1 * 1.5
a1 = 24 / 1.5
a1 = 16
So, the first term (a1) is 16 and the common ratio (r) is 1.5.
ii. Finding the 10th term:
We can use the formula for the nth term (an) to calculate the 10th term:
a10 = a1 * r^(10-1)
Substituting the values we found earlier:
a10 = 16 * 1.5^9
Calculating this:
a10 ≈ 437.4
Therefore, the 10th term of the geometric series is approximately 437.4.
iii. Finding the geometric series:
A geometric series is the sum of all terms in a G.P. To find the sum, we can use the formula:
Sn = a1 * (r^n - 1) / (r - 1)
Substituting the given values:
Sn = 16 * (1.5^10 - 1) / (1.5 - 1)
Calculating this:
Sn ≈ 10980.65
So, the sum of the geometric series is approximately 10980.65.
To find the first and common ratio of the 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 position of the term.
Given that the 2nd term is 24 and the 5th term is 81, we can write two equations:
24 = a * r^(2-1) (Equation 1)
81 = a * r^(5-1) (Equation 2)
To solve these equations, we can divide Equation 2 by Equation 1:
81/24 = a * r^(5-1) / (a * r^(2-1))
Simplifying, we get:
27/8 = r^4
Taking the fourth root of both sides, we have:
(27/8)^(1/4) = r
Simplifying further, we get:
r = 3/2
Now, we can substitute the value of r back into Equation 1 to find the first term:
24 = a * (3/2)^(2-1)
Simplifying, we get:
24 = a * 3/2
Multiplying both sides by 2/3, we find:
a = 16
Therefore, the first term (a) is 16 and the common ratio (r) is 3/2.
ii. To find the 10th term of the G.P., we can use the formula:
Tₙ = a * r^(n-1)
Substituting the values, we have:
T₁₀ = 16 * (3/2)^(10-1)
Simplifying, we get:
T₁₀ = 16 * (3/2)^9
Calculating further, we find:
T₁₀ = 16 * (19683/512)
T₁₀ = 6561/8
Therefore, the 10th term of the G.P. is 6561/8.
iii. To find the geometric series, we can write out the terms using the formula:
Tₙ = a * r^(n-1)
Substituting the values, we have:
T₁ = 16 * (3/2)^(1-1) = 16 * 1 = 16
T₂ = 16 * (3/2)^(2-1) = 16 * (3/2) = 24
T₃ = 16 * (3/2)^(3-1) = 16 * (9/4) = 36
T₄ = 16 * (3/2)^(4-1) = 16 * (27/8) = 54
T₅ = 16 * (3/2)^(5-1) = 16 * (81/16) = 81
Therefore, the geometric series is: 16, 24, 36, 54, 81.