1)Find the sum of the first eight terms of the Geometric progression 256,128,64,32
2)How many terms should be taken from the Geometric progression 4,12,36 for the sum to be 2188
1)
The sum of n numbers in Geometric progression is:
Sn=a1*[(1-q^n)/(1-q)]
Where:
a1 is first number in progresion
q is the common ratio.
In your case:
a1=32
q=2
Sn=S8=32*[(1-2^8)/81-2]
S8=32*[(1-256)/(1-2)]
S8=32*( -255)/( -1)
S8=32*255
S8=8160
2)
I am not shure that this question have solution.
Geometric progression in this case:
Six terms:
4,12,36,108,324,972
4+12+36+108+324+972=1456
Seven terms:
4,12,36,108,324,972,2916
4+12+36+108+324+972+2916=4372
In first question:
Sn=S8=32*[(1-2^8)/(1-2)]
In a geometric progression,the product of the 2nd and 4th terms is double the 5th terms and the sum of the first four terms is 80.find the gp
To find the sum of the first eight terms of a geometric progression, you can use the formula:
S_n = a * (1 - r^n) / (1 - r)
where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
Let's apply this formula to the given geometric progression:
1) First, identify the values for a, r, and n:
- a = 256 (the first term)
- r = (128 / 256) = 0.5 (common ratio)
- n = 8 (number of terms)
2) Substituting the values into the formula:
S_8 = 256 * (1 - 0.5^8) / (1 - 0.5)
= 256 * (1 - 0.00390625) / 0.5
= 256 * 0.99609375 / 0.5
= 510.1171875
Therefore, the sum of the first eight terms of the geometric progression 256, 128, 64, 32 is approximately 510.12.
To find the number of terms required for the sum to be a specific value in a geometric progression, you can use a similar approach.
1) The formula to find the sum of a geometric progression is still:
S_n = a * (1 - r^n) / (1 - r)
2) However, this time we need to find the value of n, so we can rearrange the formula:
S_n = a * (1 - r^n) / (1 - r)
(1 - r^n) / (1 - r) = S_n / a
1 - r^n = (S_n / a) * (1 - r)
r^n = 1 - ((S_n / a) * (1 - r))
n = log(r^n) / log(r)
Let's apply this formula to the second problem:
2) First, identify the values for a, r, and S_n:
- a = 4 (the first term)
- r = (12 / 4) = 3 (common ratio)
- S_n = 2188 (desired sum)
3) Substituting the values into the formula:
n = log(2188 / 4) / log(3)
≈ log(547) / log(3) ≈ 6.69
Therefore, we would need to take approximately 7 terms from the geometric progression 4, 12, 36 for the sum to be 2188.