The sum of the first and the third term of a G.P is 10 if the first term is 2 find:
a. The common ratio.
b. The 6th term
2 + 2r^2 = 10
solve for r, and then you want 2r^5
To find the common ratio, we need to use the formula for the nth term of a geometric progression (G.P.), which is:
an = ar^(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.
a. The sum of the first and the third term is given as 10, so we can write the equation:
a + ar^(2-1) = 10
Substituting the given values, we have:
2 + 2r = 10
Simplifying the equation, we get:
2r = 10 - 2
2r = 8
Dividing both sides by 2, we find:
r = 4
Therefore, the common ratio is 4.
b. To find the 6th term, we can substitute the values of the first term (a = 2), common ratio (r = 4), and the position of the term (n = 6) into the formula:
a6 = 2 * 4^(6-1)
Simplifying the equation, we get:
a6 = 2 * 4^5
a6 = 2 * 1024
a6 = 2048
Therefore, the 6th term of the G.P. is 2048.
To find the common ratio and the 6th term of a geometric progression (G.P) with the given information, we can use the formula for the nth term of a G.P.
The formula to find the nth term of a G.P is:
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.
Now, let's use the given information to find the common ratio and the 6th term of the G.P.
a. To find the common ratio:
Given: The sum of the first and the third term is 10, and the first term is 2.
The sum of the first and third term can be expressed as:
a₁ + a₃ = 2 + (2 * r^2)
Since the sum is given as 10, we can write the equation:
2 + (2 * r^2) = 10
Simplifying the equation:
2 * r^2 = 10 - 2
2 * r^2 = 8
Dividing both sides by 2:
r^2 = 4
Taking the square root of both sides:
r = ±2
Since a common ratio cannot be negative, we will take the positive value:
r = 2
Therefore, the common ratio is 2.
b. To find the 6th term:
Now that we have the common ratio, we can use it to find the 6th term:
a₆ = a₁ * r^(6-1)
Substituting the given values:
a₆ = 2 * 2^(6-1)
a₆ = 2 * 2^5
a₆ = 2 * 32
a₆ = 64
Therefore, the 6th term of the G.P is 64.