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GP1 is a geometric progression with the first term 1 and common ratio r which is a positive integer greater than 1. GP2 is another geometric progression with the first term 1 and common ratio r4. What will be the remainder when the sum of the first four terms of GP2 is divided by the first four terms of GP-1?
- a)A monomial in x
- b)0
- c)4
- d)A quadratic polynomial in x
- e)1
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
GP1is a geometric progression with the first term 1 and common ratio r...
Hence, option 3.
Most Upvoted Answer
GP1is a geometric progression with the first term 1 and common ratio r...
To find the remainder when the sum of the first four terms of GP2 is divided by the first four terms of GP1, let's first find the first four terms of both geometric progressions.
GP1:
The first term of GP1 is 1, and the common ratio is r. Therefore, the first four terms of GP1 are:
1, r, r^2, r^3
GP2:
The first term of GP2 is 1, and the common ratio is r^4. Therefore, the first four terms of GP2 are:
1, r^4, (r^4)^2 = r^8, (r^4)^3 = r^12
Now, let's find the sum of the first four terms of GP2:
1 + r^4 + r^8 + r^12
To find the remainder when this sum is divided by the first four terms of GP1, we need to divide the above expression by each term of GP1 and find the remainder for each term.
1 + r^4 + r^8 + r^12 divided by 1: Remainder = 0
1 + r^4 + r^8 + r^12 divided by r: Remainder = r^3
1 + r^4 + r^8 + r^12 divided by r^2: Remainder = r^2 + 1
1 + r^4 + r^8 + r^12 divided by r^3: Remainder = r + r^2 + 1
Therefore, the remainder when the sum of the first four terms of GP2 is divided by the first four terms of GP1 is 0, r^3, r^2 + 1, and r + r^2 + 1.
Among the given options, the remainder is 4, which matches with the remainder r^2 + 1. Hence, the correct answer is option C) 4.
GP1:
The first term of GP1 is 1, and the common ratio is r. Therefore, the first four terms of GP1 are:
1, r, r^2, r^3
GP2:
The first term of GP2 is 1, and the common ratio is r^4. Therefore, the first four terms of GP2 are:
1, r^4, (r^4)^2 = r^8, (r^4)^3 = r^12
Now, let's find the sum of the first four terms of GP2:
1 + r^4 + r^8 + r^12
To find the remainder when this sum is divided by the first four terms of GP1, we need to divide the above expression by each term of GP1 and find the remainder for each term.
1 + r^4 + r^8 + r^12 divided by 1: Remainder = 0
1 + r^4 + r^8 + r^12 divided by r: Remainder = r^3
1 + r^4 + r^8 + r^12 divided by r^2: Remainder = r^2 + 1
1 + r^4 + r^8 + r^12 divided by r^3: Remainder = r + r^2 + 1
Therefore, the remainder when the sum of the first four terms of GP2 is divided by the first four terms of GP1 is 0, r^3, r^2 + 1, and r + r^2 + 1.
Among the given options, the remainder is 4, which matches with the remainder r^2 + 1. Hence, the correct answer is option C) 4.
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GP1is a geometric progression with the first term 1 and common ratio r which is a positive integer greater than 1. GP2 is another geometric progression with the first term 1 and common ratio r4. What will be the remainder when the sum of the first four terms of GP2 is divided by the first four terms of GP-1?a)A monomial in xb)0c)4d)A quadratic polynomial in xe)1Correct answer is option 'C'. Can you explain this answer?
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