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If (1 + x + x2)n = a0 + a1x + a2x2 + ..... + a2nx2n, then a0 + a3 + a6 + ....... is equal to
- a)3n – 1
- b)2n – 1
- c)3n
- d)3n – 1
Correct answer is option 'D'. Can you explain this answer?
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If (1 + x + x2)n = a0 + a1x + a2x2 + ..... + a2nx2n, then a0 + a3 + a6...
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If (1 + x + x2)n = a0 + a1x + a2x2 + ..... + a2nx2n, then a0 + a3 + a6...
The given expression can be rewritten as:
(1 + x + x^2)^n = a0 + a1x + a2x^2 + ... + a2nx^(2n)
To find the terms a0, a3, a6, ..., we need to find the terms with powers of x that are multiples of 3.
When expanding (1 + x + x^2)^n, the powers of x will be of the form (x^k)(x^l)(x^m), where k + l + m = 2n.
If we want the power of x to be a multiple of 3, we need k, l, and m to all be multiples of 3.
Since k + l + m = 2n, and we want k, l, and m to be multiples of 3, we can let k = 3a, l = 3b, and m = 3c, where a, b, and c are integers.
Substituting these values into k + l + m = 2n, we get 3a + 3b + 3c = 2n.
Simplifying, we have a + b + c = (2n)/3.
Since a, b, and c are integers, (2n)/3 must be an integer.
Therefore, for the powers of x to be multiples of 3, n must be a multiple of 3.
So, the terms a0, a3, a6, ... are the coefficients of x^0, x^3, x^6, ..., respectively.
Since n must be a multiple of 3, we can write n as 3k, where k is an integer.
Then, the terms a0, a3, a6, ... are the coefficients of x^0, x^3, x^6, ..., respectively, when expanding (1 + x + x^2)^(3k).
Therefore, a0, a3, a6, ... = 3k.
(1 + x + x^2)^n = a0 + a1x + a2x^2 + ... + a2nx^(2n)
To find the terms a0, a3, a6, ..., we need to find the terms with powers of x that are multiples of 3.
When expanding (1 + x + x^2)^n, the powers of x will be of the form (x^k)(x^l)(x^m), where k + l + m = 2n.
If we want the power of x to be a multiple of 3, we need k, l, and m to all be multiples of 3.
Since k + l + m = 2n, and we want k, l, and m to be multiples of 3, we can let k = 3a, l = 3b, and m = 3c, where a, b, and c are integers.
Substituting these values into k + l + m = 2n, we get 3a + 3b + 3c = 2n.
Simplifying, we have a + b + c = (2n)/3.
Since a, b, and c are integers, (2n)/3 must be an integer.
Therefore, for the powers of x to be multiples of 3, n must be a multiple of 3.
So, the terms a0, a3, a6, ... are the coefficients of x^0, x^3, x^6, ..., respectively.
Since n must be a multiple of 3, we can write n as 3k, where k is an integer.
Then, the terms a0, a3, a6, ... are the coefficients of x^0, x^3, x^6, ..., respectively, when expanding (1 + x + x^2)^(3k).
Therefore, a0, a3, a6, ... = 3k.
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If (1 + x + x2)n = a0 + a1x + a2x2 + ..... + a2nx2n, then a0 + a3 + a6 + ....... is equal toa)3n – 1b)2n – 1c)3nd)3n – 1Correct answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If (1 + x + x2)n = a0 + a1x + a2x2 + ..... + a2nx2n, then a0 + a3 + a6 + ....... is equal toa)3n – 1b)2n – 1c)3nd)3n – 1Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If (1 + x + x2)n = a0 + a1x + a2x2 + ..... + a2nx2n, then a0 + a3 + a6 + ....... is equal toa)3n – 1b)2n – 1c)3nd)3n – 1Correct answer is option 'D'. Can you explain this answer?.
If (1 + x + x2)n = a0 + a1x + a2x2 + ..... + a2nx2n, then a0 + a3 + a6 + ....... is equal toa)3n – 1b)2n – 1c)3nd)3n – 1Correct answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If (1 + x + x2)n = a0 + a1x + a2x2 + ..... + a2nx2n, then a0 + a3 + a6 + ....... is equal toa)3n – 1b)2n – 1c)3nd)3n – 1Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If (1 + x + x2)n = a0 + a1x + a2x2 + ..... + a2nx2n, then a0 + a3 + a6 + ....... is equal toa)3n – 1b)2n – 1c)3nd)3n – 1Correct answer is option 'D'. Can you explain this answer?.
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