If the sum of first four terms of GP is 30, sum of last four terms of GP is 960 and first term is 2 then find common ratio.
2(r^4-1)/(r-1) = 30
r=2
The last 4 terms don't matter, unless you are interested in how many there are.
2(r^4-1)/(r-1) = 30
r=2
Cannot we say by comparing? there is no simplification? How can u say that direct answer is ?
To find the common ratio of a geometric progression (GP) given the sum of the first four terms and the sum of the last four terms, we can use the following steps:
Step 1: Understand the problem.
In this case, we are given:
- The sum of the first four terms of the GP is 30.
- The sum of the last four terms of the GP is 960.
- The first term of the GP is 2.
Step 2: Use the formula for the sum of terms in a GP.
The sum of the first n terms of a GP can be calculated using 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 of the GP.
- r is the common ratio of the GP.
Step 3: Write the equations based on the given information.
From the problem, we can write the following equations:
- For the sum of the first four terms: 30 = 2(1 - r^4) / (1 - r)
- For the sum of the last four terms: 960 = 2(1 - r^4n) / (1 - r^4)
Step 4: Solve the equations.
To find the common ratio (r), we need to solve the two equations simultaneously. However, the equations are non-linear, so we cannot easily solve them algebraically. We'll need to use numerical methods or trial and error to find the common ratio that satisfies the given conditions.
One possible approach is to use trial and error. You can start by guessing a value for r, plug it into the equations, and check if the sums of the first and last four terms match the given values. If they don't match, adjust your guess and try again until you find a value of r that satisfies both equations.
For example, let's start with a guess for r = 0.9.
- For the sum of the first four terms:
30 = 2(1 - 0.9^4) / (1 - 0.9)
30 = 2(1 - 0.9^4) / 0.1
300 = 2(1 - 0.9^4)
300 = 2(1 - 0.6561)
300 = 2(0.3439)
300 = 0.6878
- For the sum of the last four terms:
960 = 2(1 - 0.9^4n) / (1 - 0.9^4)
960 = 2(1 - 0.9^4n) / (1 - 0.6561)
960 = 2(1 - 0.9^4n) / 0.3439
960 = 5.8113(1 - 0.9^4n)
By comparing the results of the equations with the given values, we can see that our guess for r = 0.9 does not satisfy both equations. We need to repeat the process, adjusting the guess for r until we find the correct value that satisfies both equations.