for the geometric progression 20,5,1and half. find (1)the common ratio (2)the n the term (3)the sum of the first 8 term
You claim it is a GP, so the common ratio must be 5/20 = 1/4
so the 3rd term should be 5/4, which is not what you stated.
term(n) = ar^(n-1) = 20(1/4)^(n-1)
= (5)(4)(1/4)^(n-1)
= 5(4)(4)^(1-n)
= 5(4)^(2-n)
sum(8) = a(1 - r^8)/(1 - r)
= 20(1 - (1/4)^8)/(3/4)
= (80/3)(1 - 1/65536)
= (80/3)(65535/65536)
= 109225/4096 or appr 26.667
Please help me find the 7th term
To find the common ratio in a geometric progression, we need to divide any term by its preceding term.
(1) Common Ratio:
Let's take the second term, 5, and divide it by the first term, 20:
Common Ratio (r) = 5 / 20 = 1/4
Therefore, the common ratio is 1/4.
(2) nth Term Formula:
The nth term of a geometric progression can be calculated using the formula:
an = a1 * (r)^(n-1)
Where:
an = the nth term
a1 = the first term
r = the common ratio
n = the term number
Let's calculate the 5th term:
a5 = 20 * (1/4)^(5-1)
a5 = 20 * (1/4)^4
a5 = 20 * (1/4)^4
a5 = 20 * (1/256)
a5 = 20/256
a5 = 5/64
Therefore, the 5th term is 5/64.
(3) Sum of the First 8 Terms:
The sum of the first n terms in a geometric progression can be calculated using the formula:
Sn = a1 * (1 - r^n) / (1 - r)
Where:
Sn = the sum of the first n terms
a1 = the first term
r = the common ratio
n = the number of terms
Let's calculate the sum of the first 8 terms:
S8 = 20 * (1 - (1/4)^8) / (1 - 1/4)
S8 = 20 * (1 - (1/4)^8) / (3/4)
S8 = 20 * (1 - 1/65536) / (3/4)
S8 = 20 * (65535/65536) / (3/4)
S8 = 20 * 2621400/196608
S8 = 52428000/196608
S8 = 266.66667
Therefore, the sum of the first 8 terms is approximately 266.66667.
To find the common ratio (r) of a geometric progression, we can use the formula:
r = (2nd term) / (1st term)
In this case, the 2nd term is 5 and the 1st term is 20. Therefore:
r = 5 / 20 = 1/4
So, the common ratio (r) is 1/4.
To find the nth term of a geometric progression, we can use the formula:
nth term = (1st term) * (common ratio)^(n - 1)
In this case, the 1st term is 20 and the common ratio is 1/4. Let's consider the given terms to find the pattern:
20, 5, 1, 1/2
We observe that each term is obtained by dividing the previous term by 4. Hence, the pattern is dividing by 4 each time. Therefore:
2nd term = (1st term) / 4
3rd term = (2nd term) / 4
4th term = (3rd term) / 4
So, the 2nd term is 20 / 4 = 5, the 3rd term is 5 / 4 = 1, and the 4th term is 1 / 4 = 1/4.
Hence, the 4th term is the same as the last given term, which is 1/4.
To find the sum of the first 8 terms of a geometric progression, we can use the formula:
sum of first n terms = (1st term) * (1 - (common ratio)^n) / (1 - common ratio)
In this case, the 1st term is 20, the common ratio is 1/4, and we need the sum of the first 8 terms (n = 8). Plugging in these values into the formula:
sum of first 8 terms = 20 * (1 - (1/4)^8) / (1 - 1/4)
Simplifying the formula:
sum of first 8 terms = 20 * (1 - 1/65536) / (3/4)
sum of first 8 terms = 20 * (65535/65536) / (3/4)
sum of first 8 terms = (20 * 65535 * 4) / (65536 * 3)
sum of first 8 terms = (20 * 21845) / (32768)
sum of first 8 terms = 437000 / 32768
sum of first 8 terms ≈ 13.34375