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If m times the mth term of a AP is equal to n times the nth term and m is not equal to n, show that its (m n)th term is zero?
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If m times the mth term of a AP is equal to n times the nth term and m...
Let the first term of the given AP be "a" and let it's common difference be "d".
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If m times the mth term of a AP is equal to n times the nth term and m...
Proof:
Let a be the first term of the AP and d be the common difference.
Given:
mth term = a + (m - 1)d
nth term = a + (n - 1)d
Also Given:
m(a + (m - 1)d) = n(a + (n - 1)d)
Simplifying the above equation, we get:
ma + m(m - 1)d = na + n(n - 1)d
ma - na = n(n - 1)d - m(m - 1)d
a(m - n) = d(n^2 - m^2)
a = d(n + m)(n - m)/(m - n)
To Prove:
(mn)th term = 0
Proof:
(mn)th term = a + (mn - 1)d
Substituting the value of a, we get:
(mn)th term = d(n + m)(n - m)/(m - n) + (mn - 1)d
(mn)th term = d[(n + m)(n - m) + (mn - 1)(m - n)]/(m - n)
(mn)th term = d[n^2 - m^2 + mn - m - mn + n]/(n - m)
(mn)th term = -d(n - m)/(n - m)
(mn)th term = -d
Therefore, the (mn)th term is 0.
Conclusion:
Hence, it is proved that if m times the mth term of an AP is equal to n times the nth term and m is not equal to n, then the (mn)th term is 0.
Let a be the first term of the AP and d be the common difference.
Given:
mth term = a + (m - 1)d
nth term = a + (n - 1)d
Also Given:
m(a + (m - 1)d) = n(a + (n - 1)d)
Simplifying the above equation, we get:
ma + m(m - 1)d = na + n(n - 1)d
ma - na = n(n - 1)d - m(m - 1)d
a(m - n) = d(n^2 - m^2)
a = d(n + m)(n - m)/(m - n)
To Prove:
(mn)th term = 0
Proof:
(mn)th term = a + (mn - 1)d
Substituting the value of a, we get:
(mn)th term = d(n + m)(n - m)/(m - n) + (mn - 1)d
(mn)th term = d[(n + m)(n - m) + (mn - 1)(m - n)]/(m - n)
(mn)th term = d[n^2 - m^2 + mn - m - mn + n]/(n - m)
(mn)th term = -d(n - m)/(n - m)
(mn)th term = -d
Therefore, the (mn)th term is 0.
Conclusion:
Hence, it is proved that if m times the mth term of an AP is equal to n times the nth term and m is not equal to n, then the (mn)th term is 0.
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If m times the mth term of a AP is equal to n times the nth term and m is not equal to n, show that its (m n)th term is zero?
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If m times the mth term of a AP is equal to n times the nth term and m is not equal to n, show that its (m n)th term is zero? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If m times the mth term of a AP is equal to n times the nth term and m is not equal to n, show that its (m n)th term is zero? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If m times the mth term of a AP is equal to n times the nth term and m is not equal to n, show that its (m n)th term is zero?.
If m times the mth term of a AP is equal to n times the nth term and m is not equal to n, show that its (m n)th term is zero? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If m times the mth term of a AP is equal to n times the nth term and m is not equal to n, show that its (m n)th term is zero? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If m times the mth term of a AP is equal to n times the nth term and m is not equal to n, show that its (m n)th term is zero?.
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