Question

Let a1,a2,a3..... be an arithmetic progression and b1,b2,b3...... be a geometric progression. The sequence c1,c2,c3,.... is such that cn=an+bn∀n∈N. Suppose c1=1.c2=4.c3=15andc4=2. The common ratio of geometric progression is equal to

Answer 1

In this problem, we have an arithmetic progression and a geometric progression. We are given a sequence, and we need to find the common ratio of the geometric progression based on the given values. By solving the system of equations, we find that there is no consistent value for the common ratio.

Step-by-step explanation:

In this problem, we have an arithmetic progression (AP) and a geometric progression (GP). We are given a sequence, c1, c2, c3, ... which is defined as cn = an + bn for all n. We are given the values of c1, c2, c3, and c4, and we need to find the common ratio of the geometric progression (GP).

Given that c1 = 1, c2 = 4, c3 = 15, and c4 = 2, we can substitute these values into the formula cn = an + bn to get the following equations:

1. a1 + b1 = 1
2. a2 + b2 = 4
3. a3 + b3 = 15
4. a4 + b4 = 2

To solve this system of equations, we need to eliminate either the an or bn terms. Let's eliminate the an terms by subtracting equation (i) from equation (i+1):

(a2 + b2) - (a1 + b1) = 4 - 1
a2 - a1 + b2 - b1 = 3

Similarly, we can also subtract equation (i+1) from equation (i+2):

(a3 + b3) - (a2 + b2) = 15 - 4
a3 - a2 + b3 - b2 = 11

And finally, we can subtract equation (i+2) from equation (i+3):

(a4 + b4) - (a3 + b3) = 2 - 15
a4 - a3 + b4 - b3 = -13

We can simplify these equations to:

1. a2 - a1 + b2 - b1 = 3
2. a3 - a2 + b3 - b2 = 11
3. a4 - a3 + b4 - b3 = -13

Since the terms of the arithmetic progression (AP) are canceled out, we are left with terms of the geometric progression (GP):

1. b2 - b1 = 3
2. b3 - b2 = 11
3. b4 - b3 = -13

We can now further simplify these equations to:

1. b2 = b1 + 3
2. b3 = b2 + 11
3. b4 = b3 - 13

Now, let's substitute these values of b2, b3, and b4 into equation (vii) to find the common ratio of the geometric progression:

b3 = b2 + 11
b2 + 11 = b1 + 3 + 11
b2 + 11 = b1 + 14
b2 = b1 + 3

Substituting this value of b2 into equation (viii):

b4 = b3 - 13
b1 + 17 = b1 + 14 - 13
b1 + 17 = b1 + 1
17 = 1

There is a contradiction in the last equation, which means there is no consistent value for the common ratio of the geometric progression (GP). Therefore, the common ratio cannot be determined from the given information.