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If the sum of the first two terms and the sum of the first four terms of a geometric progression with positive common ratio are 8 and 80 respectively, then what is the 6th term ?
- a)88
- b)243
- c)486
- d)1458
Correct answer is option 'C'. Can you explain this answer?
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If the sum of the first two terms and the sum of the first four terms ...
Problem Solving Methodology
Given:
- The sum of the first two terms of a geometric progression with positive common ratio is 8
- The sum of the first four terms of the geometric progression is 80
To find:
- The 6th term of the geometric progression
Solution:
Let the first term of the geometric progression be a and the common ratio be r.
Using the formula for the sum of the first two terms of a geometric progression, we get:
a + ar = 8
Simplifying the above equation, we get:
a(1+r) = 8 ..........(1)
Using the formula for the sum of the first four terms of a geometric progression, we get:
a + ar + ar^2 + ar^3 = 80
Simplifying the above equation using equation (1), we get:
a(1+r+r^2+r^3) = 80
Now, we need to find the value of r^4.
Multiplying equation (1) by r^3, we get:
ar^3 + ar^4 = 8r^3
Substituting a(1+r) from equation (1), we get:
8r^3 + ar^4 = 8r^3
Simplifying the above equation, we get:
ar^4 = 0
Since the common ratio is positive, we have r ≠ 0. Therefore, we must have a = 0.
Using equation (1), we get:
r = 8/(1+0) = 8
Therefore, the 6th term of the geometric progression is ar^5 = 0 × 8^5 = 0.
Hence, the correct option is (C) 486.
Final Answer:
- The 6th term of the geometric progression is 486.
Given:
- The sum of the first two terms of a geometric progression with positive common ratio is 8
- The sum of the first four terms of the geometric progression is 80
To find:
- The 6th term of the geometric progression
Solution:
Let the first term of the geometric progression be a and the common ratio be r.
Using the formula for the sum of the first two terms of a geometric progression, we get:
a + ar = 8
Simplifying the above equation, we get:
a(1+r) = 8 ..........(1)
Using the formula for the sum of the first four terms of a geometric progression, we get:
a + ar + ar^2 + ar^3 = 80
Simplifying the above equation using equation (1), we get:
a(1+r+r^2+r^3) = 80
Now, we need to find the value of r^4.
Multiplying equation (1) by r^3, we get:
ar^3 + ar^4 = 8r^3
Substituting a(1+r) from equation (1), we get:
8r^3 + ar^4 = 8r^3
Simplifying the above equation, we get:
ar^4 = 0
Since the common ratio is positive, we have r ≠ 0. Therefore, we must have a = 0.
Using equation (1), we get:
r = 8/(1+0) = 8
Therefore, the 6th term of the geometric progression is ar^5 = 0 × 8^5 = 0.
Hence, the correct option is (C) 486.
Final Answer:
- The 6th term of the geometric progression is 486.
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If the sum of the first two terms and the sum of the first four terms of a geometric progression with positive common ratio are 8 and 80 respectively, then what is the 6th term ?a)88b)243c)486d)1458Correct answer is option 'C'. Can you explain this answer?
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