The ratio of the forth and 12th terms of a g.p. with positive common ratio is 1:256. If the difference of two terms be 61.2. Find the sum of 8 term of this series.

ar^3/ar^11 = 1/256

1/r^8 = 1/256
r = 2

Now you can find a, so the sum

S8 = a(2^8-1)/(2-1) = 127a

Let's assume the first term of the geometric progression is a and the common ratio is r.

The ratio of the fourth term to the twelfth term is given as 1:256. So we can write:

a * r^3 / (a * r^11) = 1 / 256

Simplifying this equation gives us:

r^8 = 1 / 256

Taking the square root of both sides gives us:

r^4 = 1 / 16

r = (1/16)^(1/4)

r = 1/2

Now, we can find the value of a using the difference of the two terms. We know that the difference is 61.2, so:

a * (r^11 - r^3) = 61.2

Plugging in the value of r, we get:

a * ((1/2)^11 - (1/2)^3) = 61.2

a * (1/2048 - 1/8) = 61.2

a * (1/2048 - 256/2048) = 61.2

a * (-255/2048) = 61.2

a = -61.2 * (2048/255)

a = -492.48

Now that we have the first term (a = -492.48) and the common ratio (r = 1/2), we can find the sum of the first 8 terms using the formula for the sum of a geometric progression:

Sum of n terms = a * (1 - r^n) / (1 - r)

Substituting in the values, we get:

Sum of 8 terms = -492.48 * (1 - (1/2)^8) / (1 - 1/2)

Simplifying this expression gives us:

Sum of 8 terms = -492.48 * (1 - 1/256) / (1/2)

Sum of 8 terms = -492.48 * (255/256) / (1/2)

Sum of 8 terms = -492.48 * (255/256) * 2

Sum of 8 terms = -492.48 * 255

Sum of 8 terms ≈ - 125,536.4

Therefore, the sum of the first 8 terms of the geometric progression is approximately -125,536.4.

To solve this problem, we need to understand and use the formulae for the terms of a geometric progression (g.p.), as well as the formula for the sum of a geometric series.

Let's denote the first term of the g.p. as 'a', and the common ratio as 'r'.

The ratio of the fourth term to the 12th term is given as 1:256. This means that:

(a * r^3) / (a * r^11) = 1/256

To simplify, we can cancel out the 'a' terms:

r^3 / r^11 = 1/256

Using the property of exponents, we subtract the exponents:

r^(3-11) = 1/256

r^(-8) = 1/256

To further simplify, we can express 1/256 as a power of 2:

r^(-8) = 2^(-8)

In order to compare the exponents, we can rewrite the left side of the equation as a positive exponent:

1 / (r^8) = 2^(-8)

Taking the reciprocal of both sides, we have:

r^8 = 2^8

Now, we can solve for the value of 'r':

r = √(2^8)

r = 2^4

r = 16

Now that we know the common ratio 'r' is 16, we can find the value of 'a' using the given difference of two terms.

The difference between two terms is given as 61.2, so we have:

a * (r - 1) = 61.2

a * (16 - 1) = 61.2

15a = 61.2

a = 61.2 / 15

a ≈ 4.08

We now have the first term 'a' and the common ratio 'r' of the g.p. With this information, we can find the sum of the eight terms of the series using the formula for the sum of a geometric series:

Sum = a * (r^n - 1) / (r - 1)

Substituting the values:

Sum = 4.08 * (16^8 - 1) / (16 - 1)

Sum ≈ 4.08 * (4294967296 - 1) / 15

Sum ≈ 4.08 * 4294967295 / 15

Sum ≈ 1,112,134,973.4

Therefore, the sum of the eight terms of the geometric series is approximately 1,112,134,973.4.

a=0.03