The third,fourth and eight term of A.P form first three consecutive term of G.P.if the sum of the first ten term of A.P is 85.calculate the first term of both A.P and G.P,common ratio and sum of 5 term of G.P
To find the first term of the arithmetic progression (AP) and the geometric progression (GP), as well as the common ratio and sum of 5 terms in the GP, let's break down the given information step by step:
Step 1: Finding the first term of the AP
We are given that the third, fourth, and eighth terms of the AP form the first three consecutive terms of a GP. Let's assume that the common difference of the AP is 'd', and the first term of the AP is 'a'.
The third term will be: a + 2d
The fourth term will be: a + 3d
The eighth term will be: a + 7d
Since the first three terms of the AP form a GP, we can express the terms in terms of the common ratio (r):
(a + 3d) / (a + 2d) = (a + 7d) / (a + 3d)
To solve this equation, we cross-multiply:
(a + 3d) * (a + 3d) = (a + 2d) * (a + 7d)
a^2 + 6ad + 9d^2 = a^2 + 9ad + 14d^2
3ad - 5d^2 = 0
d(3a - 5d) = 0
Since the value of 'd' cannot be zero (as it is the common difference), we can ignore the second term.
So, 3a - 5d = 0
3a = 5d
Given that the sum of the first ten terms of the AP is 85, we can use the formula for the sum of an AP to establish another equation:
Sum = (n/2)(2a + (n-1)d)
Substituting the given values:
85 = (10/2)(2a + (10-1)d)
85 = 5(2a + 9d)
17 = 2a + 9d
Now, we have two equations:
3a = 5d
2a + 9d = 17
Step 2: Solving the equations
We can solve these two equations simultaneously to find the values of 'a' and 'd'.
From the first equation, we can express 'a' in terms of 'd':
3a = 5d
a = (5d) / 3
Substituting this value of 'a' into the second equation:
2((5d) / 3) + 9d = 17
(10d / 3) + 9d = 17
(10d + 27d) / 3 = 17
37d = 51
d = 51 / 37
Dividing both the numerator and denominator by the greatest common divisor (GCD), which is 1 in this case, we get:
d = 1.378
Now, substituting this value of 'd' back into the first equation:
3a = 5(1.378)
3a = 6.89
a = 6.89 / 3
a = 2.296
Therefore, the first term of the AP is a = 2.296.
Step 3: Finding the common ratio of the GP
Since the first three terms of the AP form the first three consecutive terms of the GP, we can find the common ratio by dividing the fourth term by the third term:
Common Ratio (r) = (a + 3d) / (a + 2d)
Substituting the values:
r = (2.296 + 3(1.378)) / (2.296 + 2(1.378))
r = 2.296 + 4.134 / 2.296 + 2.756
r = 6.43 / 5.052
r = 1.273
Therefore, the common ratio of the GP is r = 1.273.
Step 4: Finding the sum of 5 terms in the GP
To find the sum of the first five terms of the GP, we can use the formula:
Sum = a * (1 - r^n) / (1 - r)
In this case, a is the first term of the GP, which is a + 3d:
Sum = (a + 3d) * (1 - r^5) / (1 - r)
Substituting the values:
Sum = (2.296 + 3(1.378)) * (1 - (1.273)^5) / (1 - 1.273)
Sum = 2.296 + 4.134 * -0.250 / -0.273
Sum = 2.296 - 1.033
Sum = 1.263
Therefore, the sum of the first five terms in the GP is 1.263.
To summarize:
- The first term of the AP is 2.296.
- The common ratio of the GP is 1.273.
- The sum of the first five terms in the GP is 1.263.
just write it as math, and then solve. If the AP has 1st term a, and the GP has first term g, then
a+2d = g
a+3d = gr
a+7d = gr^2
10/2 (2a+d) = 85