If the 2nd term of a geometric progression is 6 and 4th term is 54 ,find the:(1)1st term(2)5th term
To answer
To find the first term and the fifth term of a geometric progression given the second term (t2 = 6) and the fourth term (t4 = 54), we can use the formula for the nth term of a geometric progression:
tn = ar^(n-1)
Where a is the first term, r is the common ratio, and n is the term number.
First, we need to find the common ratio (r). We can do this by dividing the fourth term by the second term:
r = t4 / t2 = 54 / 6 = 9
Now that we have the common ratio (r = 9), we can find the first term (a) using the second term (t2 = 6):
t2 = ar^(2-1)
6 = a * 9^1
6 = 9a
Solving for a, we get:
a = 6 / 9 = 2/3 or 0.6667 (rounded to four decimal places)
Now, we can find the fifth term (t5) using the first term (a = 2/3) and the common ratio (r = 9):
t5 = a * r^(5-1)
t5 = (2/3) * 9^4
Calculating this, we get:
t5 = (2/3) * 6561 = 4374
Therefore, the answers are:
(1) The first term (a) is 2/3 or approximately 0.6667.
(2) The fifth term (t5) is 4374.
To find the first term and the fifth term of a geometric progression, we need to find the common ratio, denoted by 'r', and then use it to calculate the terms.
Let's consider the given information:
The second term (a2) of the geometric progression is 6.
The fourth term (a4) is 54.
Using the formula for a geometric progression, we can write the terms as follows:
a2 = a1 * r
a4 = a1 * r^3
Now, let's substitute the given values into the equations:
6 = a1 * r ...(1)
54 = a1 * r^3 ...(2)
To solve this system of equations, we can divide equation (2) by equation (1):
(54/6) = (a1 * r^3) / (a1 * r)
9 = r^2
Taking the square root of both sides, we get:
r = ±3
Now, we have two possible values for the common ratio: r = 3 or r = -3.
To find the first term (a1), we can substitute the value of r into equation (1):
6 = a1 * 3
a1 = 2
Therefore, the first term (a1) of the geometric progression is 2.
To find the fifth term (a5), we can substitute the value of r into equation (1) and calculate accordingly:
a5 = a1 * r^4
If r = 3:
a5 = 2 * 3^4
a5 = 2 * 81
a5 = 162
If r = -3:
a5 = 2 * (-3)^4
a5 = 2 * 81
a5 = -162
Therefore, the fifth term (a5) of the geometric progression can either be 162 or -162, depending on the value of the common ratio.
You have r^2 = 54/6 = 9
so r = ±3
a1 = a2/r = ____
a5 = a4 * r = ____