If the 2nd term of a geometri progression is 6 and 4Th term is 54 find the 1st term and 5Th term
Just apply the given data to our definitions:
ar = 6
ar^3 = 54
divide the 2nd by the first
ar^3/(ar) = 54/6
r^2 = 9
r = ± 3
go back to ar = 6
if r = 3, 3a = 6, a = 2
if r = -3, -3a = 6, a = -2
5th term = ar^4 = ..... , remember there are 2 cases.
Well, well, well! Looks like we've got ourselves a geometric progression puzzle here! Let's put on our math clown shoes and solve it!
To find the first term (a) of the geometric progression, we can use the formula:
aₙ = a₁ * r^(n-1)
where aₙ represents the nth term, a₁ is the first term, r is the common ratio, and n is the position of the term.
We're given that the 2nd term (a₂) is equal to 6, so we can plug these values into the formula:
6 = a₁ * r^(2-1)
6 = a₁ * r
We're also given that the 4th term (a₄) is equal to 54:
54 = a₁ * r^(4-1)
54 = a₁ * r^3
Now, we've got ourselves a little system of equations:
6 = a₁ * r
54 = a₁ * r^3
To solve this math circus, we'll divide the second equation by the first equation:
54 / 6 = (a₁ * r^3) / (a₁ * r)
9 = r^2
Now, we can solve for r:
√9 = √(r^2)
3 = r
Since we know the value of r, we can substitute it back into one of the original equations to find the first term (a₁):
6 = a₁ * 3
a₁ = 2
Ta-da! The first term (a₁) of the geometric progression is 2!
Now, to find the fifth term, we can use the same formula with n = 5:
a₅ = a₁ * r^(5-1)
a₅ = 2 * 3^(5-1)
a₅ = 2 * 81
a₅ = 162
Voila! The fifth term (a₅) of the geometric progression is 162.
I hope my clown math skills were up to par for you!
To find the first term and fifth term of a geometric progression, we can make use of the formula for the nth term of a geometric sequence:
an = a1 * r^(n-1),
where an is the nth term, a1 is the first term, r is the common ratio, and n is the position of the term.
First, let's find the common ratio (r) of the sequence using the given information:
Second term, a2 = 6
Fourth term, a4 = 54
We can set up the following equations using the formula:
a2 = a1 * r^(2-1)
6 = a1 * r
a4 = a1 * r^(4-1)
54 = a1 * r^3
Divide the equation for a4 by a2 to eliminate a1:
54 / 6 = (a1 * r^3) / (a1 * r)
9 = r^2
Taking the square root of both sides, we find:
r = ±3
Now, let's consider each case individually:
Case 1: r = 3
Substituting r = 3 into the equation a2 = a1 * r, we can solve for a1:
6 = a1 * 3
a1 = 6 / 3
a1 = 2
To find the fifth term, we can use the formula:
a5 = a1 * r^(5-1)
a5 = 2 * 3^4
a5 = 2 * 81
a5 = 162
Therefore, in this case, the first term (a1) is 2 and the fifth term (a5) is 162.
Case 2: r = -3
Substituting r = -3 into the equation a2 = a1 * r, we can solve for a1:
6 = a1 * (-3)
a1 = 6 / (-3)
a1 = -2
To find the fifth term, we can use the formula:
a5 = a1 * r^(5-1)
a5 = -2 * (-3)^4
a5 = -2 * 81
a5 = -162
Therefore, in this case, the first term (a1) is -2 and the fifth term (a5) is -162.
In conclusion, depending on the value of the common ratio, the first term could be 2 or -2, and the fifth term could be 162 or -162.
To find the first term and fifth term of a geometric progression, we need to determine the common ratio first.
In a geometric progression, each term is found by multiplying the previous term by a constant value called the common ratio (r).
Given:
The second term (a2) is 6, and the fourth term (a4) is 54.
Let's use the formula for the nth term of a geometric progression:
an = a1 * r^(n-1)
Since we know the second term (a2), we can write it as:
a2 = a1 * r^(2-1)
6 = a1 * r
Similarly, for the fourth term (a4):
a4 = a1 * r^(4-1)
54 = a1 * r^3
Now we have two equations:
Equation 1: 6 = a1 * r
Equation 2: 54 = a1 * r^3
Divide equation 2 by equation 1 to eliminate a1:
54/6 = (a1 * r^3) / (a1 * r)
9 = r^2
Take the square root of both sides to solve for r:
r = ±3
Now we have two possible values for the common ratio, which are 3 and -3.
Using equation 1 (6 = a1 * r), substitute the values of r:
For r = 3:
6 = a1 * 3
a1 = 2
For r = -3:
6 = a1 * (-3)
a1 = -2
Hence, we have two possible first terms: 2 and -2.
Now, let's find the fifth term (a5) for each case:
For a1 = 2 and r = 3:
a5 = a1 * r^(5-1)
a5 = 2 * 3^4
a5 = 2 * 81
a5 = 162
For a1 = -2 and r = 3:
a5 = a1 * r^(5-1)
a5 = -2 * 3^4
a5 = -2 * 81
a5 = -162
Therefore, for a common ratio (r) of 3, the first term is 2 and the fifth term is 162.
And for a common ratio of -3, the first term is -2 and the fifth term is -162.