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For a positive integer n, the quadratic equation x(x+1)+(x+1)(x+2)+......+(x+n−1)(x+n)=10 n has two consecutive integral solutions, then n is equal to.
- a)12
- b)9
- c)10
- d)11
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
For a positive integern, the quadratic equationx(x+1)+(x+1)(x+2)+........
Most Upvoted Answer
For a positive integern, the quadratic equationx(x+1)+(x+1)(x+2)+........
Explanation:
Given quadratic equation:
x(x+1) + (x+1)(x+2) + ... + (x+n-1)(x+n) = 10n
Two consecutive integral solutions:
Let the two consecutive integral solutions be m and m+1.
Substitute m and m+1 into the equation:
m(m+1) + (m+1)(m+2) + ... + (m+n-1)(m+n) = 10n
(m+1)(m+2) + (m+2)(m+3) + ... + (m+n)(m+n+1) = 10n
Subtract the two equations:
[m(m+1) + (m+1)(m+2) + ... + (m+n-1)(m+n)] - [(m+1)(m+2) + (m+2)(m+3) + ... + (m+n)(m+n+1)] = 0
This simplifies to: m(m+1) - (m+n)(m+n+1) = 0
Expand and simplify:
m^2 + m - (m^2 + 2mn + n^2 + m + n) = 0
m^2 + m - m^2 - 2mn - n^2 - m - n = 0
- 2mn - n^2 - n = 0
n(2m + n + 1) = 0
Since n is positive:
2m + n + 1 = 0
n = -2m - 1
Since n is an integer:
-2m - 1 must be an integer
This implies that m must be an integer as well.
Therefore, the only value of n that satisfies the condition of having two consecutive integral solutions is when:
n = 11
Hence, the correct answer is option D (11).
Given quadratic equation:
x(x+1) + (x+1)(x+2) + ... + (x+n-1)(x+n) = 10n
Two consecutive integral solutions:
Let the two consecutive integral solutions be m and m+1.
Substitute m and m+1 into the equation:
m(m+1) + (m+1)(m+2) + ... + (m+n-1)(m+n) = 10n
(m+1)(m+2) + (m+2)(m+3) + ... + (m+n)(m+n+1) = 10n
Subtract the two equations:
[m(m+1) + (m+1)(m+2) + ... + (m+n-1)(m+n)] - [(m+1)(m+2) + (m+2)(m+3) + ... + (m+n)(m+n+1)] = 0
This simplifies to: m(m+1) - (m+n)(m+n+1) = 0
Expand and simplify:
m^2 + m - (m^2 + 2mn + n^2 + m + n) = 0
m^2 + m - m^2 - 2mn - n^2 - m - n = 0
- 2mn - n^2 - n = 0
n(2m + n + 1) = 0
Since n is positive:
2m + n + 1 = 0
n = -2m - 1
Since n is an integer:
-2m - 1 must be an integer
This implies that m must be an integer as well.
Therefore, the only value of n that satisfies the condition of having two consecutive integral solutions is when:
n = 11
Hence, the correct answer is option D (11).
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