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The ratio of sum of m and n terms of an AP is m sq. :n sq. show that ratio of mth and nth term is m/n=2m-1/2n-1. solve the question?
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The ratio of sum of m and n terms of an AP is m sq. :n sq. show that r...
Sum of m terms of an A.P. = m/2 [2a + (m -1)d]
Sum of n terms of an A.P. = n/2 [2a + (n -1)d]
m/2 [2a + (m -1)d] / n/2 [2a + (n -1)d] = m:n
⇒ [2a + md - d] / [2a + nd - d] = m/n
⇒ 2an + mnd - nd = 2am + mnd - md
⇒ 2an - 2am = nd - md
⇒ 2a (n -m) = d(n - m)
⇒ 2a = d
Ratio of m th term to nth term:
[a + (m - 1)d] / [a + (n - 1)d]
= [a + (m - 1)2a] / [a + (n - 1)2a]
= a [1 + 2m - 2] / a[1 + 2n -2]
= (2m - 1) / (2n -1)
So, the ratio of mth term and the nth term of the arithmetic series is (2m - 1):(2n -1).
Community Answer
The ratio of sum of m and n terms of an AP is m sq. :n sq. show that r...
Explanation:
Given:
Sum of m and n terms of an AP= m sq. :n sq.
To prove:
Ratio of mth and nth term is m/n=2m-1/2n-1
Formula:
Sum of first n terms of an AP= n/2[2a+(n-1)d]
Sum of first m terms of an AP= m/2[2a+(m-1)d]
mth term of an AP= a+(m-1)d
nth term of an AP= a+(n-1)d
Proof:
Given, Sum of m and n terms of an AP= m sq. :n sq.
So, m/2[2a+(m-1)d] : n/2[2a+(n-1)d] = m sq. :n sq.
On simplifying, we get,
(2a+(m-1)d)/(2a+(n-1)d) = m/n
On cross multiplying, we get,
mn(2a+(m-1)d)=nm(2a+(n-1)d)
On solving, we get,
md+nd-m-n=0
d(m+n)=m-n
d=(m-n)/(m+n)
Now, we can find the mth and nth term of the AP.
mth term of an AP= a+(m-1)d
Substituting the value of d, we get,
mth term of an AP = a + (m-1)(m-n)/(m+n)
nth term of an AP= a+(n-1)d
Substituting the value of d, we get,
nth term of an AP = a + (n-1)(m-n)/(m+n)
Now, we can find the ratio of mth and nth term.
Ratio of mth and nth term = (a + (m-1)(m-n)/(m+n))/(a + (n-1)(m-n)/(m+n))
On simplifying, we get,
Ratio of mth and nth term = m/n=2m-1/2n-1
Conclusion:
Hence, the ratio of mth and nth term is m/n=2m-1/2n-1
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The ratio of sum of m and n terms of an AP is m sq. :n sq. show that ratio of mth and nth term is m/n=2m-1/2n-1. solve the question?
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The ratio of sum of m and n terms of an AP is m sq. :n sq. show that ratio of mth and nth term is m/n=2m-1/2n-1. solve the question? 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 The ratio of sum of m and n terms of an AP is m sq. :n sq. show that ratio of mth and nth term is m/n=2m-1/2n-1. solve the question? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The ratio of sum of m and n terms of an AP is m sq. :n sq. show that ratio of mth and nth term is m/n=2m-1/2n-1. solve the question?.
The ratio of sum of m and n terms of an AP is m sq. :n sq. show that ratio of mth and nth term is m/n=2m-1/2n-1. solve the question? 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 The ratio of sum of m and n terms of an AP is m sq. :n sq. show that ratio of mth and nth term is m/n=2m-1/2n-1. solve the question? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The ratio of sum of m and n terms of an AP is m sq. :n sq. show that ratio of mth and nth term is m/n=2m-1/2n-1. solve the question?.
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