If the fifth term of a g.p. is 81 and second term is 24, find the g.p.
ar^4 = 81
ar = 24
r^3 = 81/24 = 27/8
r = 3/2
Now find a.
To find the geometric progression (g.p.), we need to determine the common ratio (r) of the sequence.
The general formula for the nth term (Tn) of a geometric progression is:
Tn = a * r^(n-1),
where:
Tn is the nth term,
a is the first term, and
r is the common ratio.
Let's use the given information to solve for the common ratio (r).
Given:
The second term (T2) is 24.
The fifth term (T5) is 81.
Using the formula Tn = a * r^(n-1), we can set up the following equations:
T2 = a * r^(2-1) ⟶ 24 = a * r,
T5 = a * r^(5-1) ⟶ 81 = a * r^4.
Now we can form a ratio using the two equations:
T5 / T2 = (a * r^4) / (a * r) ⟶ 81 / 24 = r^3.
Simplifying further:
(3^4) / (2^3) = r^3,
81 / 8 = r^3.
To solve for r, we need to take the cube root of both sides:
r = ∛(81 / 8).
Evaluating the expression ∛(81 / 8):
r ≈ 1.5.
Now that we know the common ratio (r = 1.5), we can find the first term (a) by using the second term (T2 = 24) and the formula T2 = a * r:
24 = a * 1.5.
To solve for a, we divide both sides by 1.5:
a = 24 / 1.5.
Evaluating the expression 24 / 1.5:
a = 16.
Therefore, the geometric progression (g.p.) is: 16, 24, 36, 54, 81.