In a G.P the first term is 7, the last time is 448 and the sum to n terms is 889. The common ratio and the number of terms are
To find the common ratio, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r)
where S is the sum of the series, a is the first term, and r is the common ratio.
In this case, S = 889 and a = 7. So we have:
889 = 7 / (1 - r)
Simplifying, we get:
1 - r = 7 / 889
r = 1 - (7 / 889) = 1 - 0.0079 ≈ 0.9921
To find the number of terms, we can use the formula for the sum of a finite geometric series:
S = a * (1 - r^n) / (1 - r)
where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.
In this case, S = 889, a = 7, and r ≈ 0.9921. Substituting these values, we get:
889 = 7 * (1 - 0.9921^n) / (1 - 0.9921)
1 - 0.9921^n = (1 - 0.9921) / 7
0.9921^n = 1 - (1 - 0.9921) / 7
0.9921^n = 1 - 0.0079 / 7
0.9921^n ≈ 0.9989
Taking the logarithm of both sides, we get:
n * ln(0.9921) ≈ ln(0.9989)
n ≈ ln(0.9989) / ln(0.9921)
n ≈ 456.23
Since the number of terms must be a whole number, we round n down to 456.
Therefore, the common ratio is approximately 0.9921 and the number of terms is 456.
To find the common ratio and the number of terms in a geometric progression (G.P.), we can use the following formulas:
Last term (L) = First term (A) * (common ratio)^n-1
Sum of n terms (S) = (First term (A) * (1 - (common ratio)^n)) / (1 - common ratio)
Given:
First term (A) = 7
Last term (L) = 448
Sum of n terms (S) = 889
Let's find the common ratio first:
448 = 7 * (common ratio)^(n - 1) ---(equation 1)
Next, let's use the sum formula:
889 = (7 * (1 - (common ratio)^n)) / (1 - common ratio) ---(equation 2)
Now, we can solve these equations simultaneously to find the common ratio (r) and the number of terms (n).
First, let's solve equation 1 for (common ratio)^n-1:
(common ratio)^(n - 1) = 448/7
(common ratio)^(n - 1) = 64
Taking the log base 10 of both sides, we get:
(n - 1) log(common ratio) = log(64)
(n - 1) log(common ratio) = 1.806
Next, let's solve equation 2 for (1 - (common ratio)^n):
889(1 - common ratio) = 7(1 - (common ratio)^n)
889 - 889 common ratio = 7 - 7(common ratio)^n
889 - 7 = (889 - 7)common ratio^n
882 = 882(common ratio)^n
(common ratio)^n = 1
(n log(common ratio)) = 0
n log(common ratio) = 0
Now, we have a system of equations:
(n - 1) log(common ratio) = 1.806
n log(common ratio) = 0
Dividing both equations by log(common ratio):
n - 1 = 1.806 / log(common ratio)
n = 1.806 / log(common ratio) + 1
n = (1.806 + log(common ratio)) / log(common ratio) ---(equation 3)
Since n log(common ratio) = 0, we can conclude:
(1 + log(common ratio)) / log(common ratio) = 0
1 + log(common ratio) = 0
log(common ratio) = -1
(common ratio) = 10^(-1)
(common ratio) = 0.1
Now, substituting the value of (common ratio) into equation 3:
n = (1.806 + log(0.1)) / log(0.1)
n = (1.806 - 1) / (-1)
n = 0.806 / -1
n = -0.806
However, the number of terms (n) cannot be negative. Hence, there is no valid solution for the number of terms.
Therefore, the given information is not consistent with a geometric progression.
To find the common ratio and the number of terms in a geometric progression (G.P), we can use the formulas for the nth term and the sum of n terms.
First, let's determine the common ratio (r):
The formula for the nth term of a G.P is given by:
an = a1 * r^(n-1)
where:
an = value of the nth term
a1 = first term
r = common ratio
n = number of terms
In this case, the first term (a1) is 7, and the last term is given as 448. So we have:
7 * r^(n-1) = 448
Next, let's determine the number of terms (n):
The formula for the sum of n terms of a G.P is given by:
Sn = a1 * (1 - r^n) / (1 - r)
where:
Sn = sum of the first n terms
a1 = first term
r = common ratio
n = number of terms
In this case, the sum of the first n terms (Sn) is given as 889. So we have:
889 = 7 * (1 - r^n) / (1 - r)
Now, we have two equations with two variables (r and n). We can solve these equations simultaneously to find the values of r and n.
Equation 1: 7 * r^(n-1) = 448
Equation 2: 889 = 7 * (1 - r^n) / (1 - r)
First, rearrange Equation 1 to solve for n:
r^(n-1) = 448/7
r^(n-1) = 64
Taking the logarithm of both sides (base r), we have:
(n-1) = log_r(64)
(n-1) = log(64) / log(r)
Now, substitute (n-1) into Equation 2:
889 = 7 * (1 - r^((n-1)+1)) / (1 - r)
Simplify:
889 = 7 * (1 - r^n) / (1 - r)
Multiply both sides by (1 - r) to eliminate the denominator:
889 * (1 - r) = 7 * (1 - r^n)
Expand:
889 - 889r = 7 - 7r^n
Rearrange:
7r^n - 889r + 882 = 0
Now, substituting the value of (n-1) from the previous equation, we can solve for r using numerical methods such as the Newton-Raphson method or a calculator.
Once we have the value of r, we can substitute it back into any of the previous equations to solve for n.
Alternatively, you can use an online G.P calculator or software that allows you to input the given values (first term, last term, and sum to n terms) to directly find the common ratio and the number of terms.