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
a = 1st term A.P. ... d = common difference
10 a + 45 d = 85 ... 2 a + 9 d = 17
(3a + 3d) / (4a + 6d) = (4a + 6d) / (8a + 28d)
two equations and two unknowns
Helpful for the Answer on qn above
To solve this problem, we need to understand the relationship between an arithmetic progression (A.P) and a geometric progression (G.P). Let's break down the information we have:
1. The third, fourth, and eighth terms of the A.P form the first three consecutive terms of the G.P.
2. The sum of the first ten terms of the A.P is 85.
To find the solution, let's use the formulas for both an A.P and a G.P and solve the problem step by step:
Step 1: Identifying the A.P and G.P Terms
Let's assume that the A.P has a first term 'a' and a common difference 'd'. The third term of the A.P would then be 'a + 2d', the fourth term would be 'a + 3d' and the eighth term would be 'a + 7d'.
Now, if we consider the G.P formed by these terms, the first term of the G.P would be 'a + 2d', the second term would be 'a + 3d', and the third term would be 'a + 7d'.
Step 2: Relating the A.P and G.P
Since the first three terms of the G.P are consecutive terms of the A.P, we can say that the common ratio 'r' of the G.P is equal to the ratio of the second term to the first term: '(a + 3d) / (a + 2d)'.
Therefore, we have the equation: '(a + 3d) / (a + 2d) = r'.
Step 3: Calculating the Sum of the A.P
The sum of the first ten terms of an A.P is given by the formula: 'S = (n/2) * (2a + (n-1)d)', where 'n' is the number of terms in the A.P.
Given that the sum is 85, we have the equation: (10/2) * (2a + 9d) = 85.
Step 4: Solving the Equations
Now we have two equations:
(1) (a + 3d) / (a + 2d) = r
(2) (10/2) * (2a + 9d) = 85
By solving these equations simultaneously, we can find the values of 'a', 'd', and 'r'.
Step 5: Solving for 'a', 'd', and 'r'
We will solve the equations using the method of substitution. First, let's solve equation (1) for 'r':
(a + 3d) / (a + 2d) = r
Cross-multiplying, we get:
(a + 3d) * (a + 2d) = r * (a + 2d)
Expanding and simplifying, we have:
a^2 + 5ad + 6d^2 = ra + 2rd
Moving everything to one side, we get:
a^2 + (5d - r)a + (6d^2 - 2rd) = 0
Now, let's solve equation (2) for 'a':
(10/2) * (2a + 9d) = 85
5 * (2a + 9d) = 85
10a + 45d = 85
2a + 9d = 17
We have two equations:
a^2 + (5d - r)a + (6d^2 - 2rd) = 0
2a + 9d = 17
By solving these equations, the values of 'a', 'd', and 'r' can be found.
To calculate the sum of the first 5 terms of the G.P, we can use the formula: 'S = a * (1 - r^n) / (1 - r)', where 'n' is the number of terms.
Substitute the known values of 'a' and 'r' into this formula to find the answer.