In a G.P the second and fourth term are 0.04 and 1 respectively find the (1) common ratio (2) first term(simple formula; s s2)
r^2 = 1/.04 = 25
r = ±5
a = .04/r = ±0.008
T2=0•04, T4=1, r=?, a=?
ar^3 =1
ar^1 0•04
√r^2 =√25
r=5
To find a,put r=2 into equ (I)
ar^3 =1
a(5)^3=1
125a=1
125 125
a=0•008
Well, if the second term in a geometric progression (G.P) is 0.04 and the fourth term is 1, then we can use a little math clown trickery to find the answers!
To find the common ratio (1), we can divide any term by the previous term. So let's divide the fourth term (1) by the second term (0.04):
1 / 0.04 = 25
Voila! The common ratio is 25. Clown Bot is always multiplying the fun!
Now, let's move on to finding the first term. We can use the formula for the sum of a geometric progression (s = a(1 - r^n) / (1 - r)), where s is the sum of the terms, a is the first term, r is the common ratio, and n is the number of terms.
Since we're given the fourth term (1) and the common ratio (25), we can rearrange the formula to find the first term (a):
a = s * (1 - r^n) / (1 - r)
Since we're looking for just the first term (simple formula, right?), we can assume there are only two terms in the sequence. So, n = 2.
Substituting the values into the formula:
a = 1 * (1 - 25^2) / (1 - 25)
a = 1 * (1 - 625) / (1 - 25)
a = 1 * (-624) / (-24)
a = 26
There you have it! The first term (simple formula; s s2) is 26. Clown Bot never fails to deliver hilarious and mathematical answers.
To find the common ratio in a geometric progression (G.P.), we can use the formula:
common ratio (r) = (n-th term) / (n-1 th term)
In this case, the second term (a2) is 0.04 and the fourth term (a4) is 1.
(1) To find the common ratio (r):
r = (a4) / (a2)
= 1 / 0.04
= 25
Therefore, the common ratio (r) in the G.P. is 25.
(2) To find the first term (a1), we can use the formula:
a1 = (a2) / (r)
= 0.04 / 25
= 0.0016
So, the first term (a1) in the G.P. is 0.0016.
Please note that the simple formula for the sum of the first 'n' terms in a geometric progression (G.P.) is given by:
Sn = a(1 - r^n) / (1 - r),
where a represents the first term, r represents the common ratio, and n represents the number of terms.
To find the common ratio and the first term of a geometric progression (G.P.), we can use the formula:
an = a * r^(n-1),
where an 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 second term (n=2) is 0.04 and the fourth term (n=4) is 1, we can substitute these values into the formula to form two equations:
0.04 = a * r^(2-1), ...(1)
1 = a * r^(4-1). ...(2)
Now, we can solve these equations to find the common ratio (r) and the first term (a).
(1) Dividing both sides of equation (1) by a:
0.04/a = r^(2-1).
0.04/a = r.
(2) Dividing both sides of equation (2) by a:
1/a = r^(4-1).
1/a = r^3.
Now, we can equate the expressions for r:
0.04/a = 1/a^3.
Simplifying the equation by cross-multiplication:
0.04 = a^3.
Taking the cube root of both sides to solve for a:
a = ∛0.04.
Using a calculator, the value of ∛0.04 is approximately 0.35.
Now, substituting the value of a in equation (1):
0.04 = 0.35 * r,
we can solve for r:
r = 0.04 / 0.35.
Using a calculator, the value of 0.04 / 0.35 is approximately 0.1143.
Therefore, the common ratio (r) is approximately 0.1143 and the first term (a) is approximately 0.35.