Consider the sequence defined by ak=1/k2-k for k ≥ 1. Given that aM + aM + 1 + …………+ aN – 1=1/29 for positive integers M and N with M < N. Find M+N/28
Answers
Answer:
k=29
Step-by-step explanation:
Question 1) :- Consider the sequence defined by ak = 1/(k²- k) for k ≥ 1. Given that aM + a(M + 1) + _____ + a(N – 1) = 1/29 for positive integers M and N with M < N. Find (M+N)/28 ?
Solution :-
→ ak = 1/(k² - k)
→ ak = 1/k(k - 1)
→ ak = 1/(k - 1) - 1/k [since , (k - k + 1) /k(k - 1) = 1/k(k - 1)]
so,
→ a(M) + a(M + 1) + _______ a(N - 1) = 1/29
→ {(1/(M - 1) - 1/M} + {1/M - 1/(M + 1)} + ________ {1/(N - 2) - 1/(N - 1)} = 1/29
→ 1/(M - 1) - 1/(N - 1) = 1/29
→ (N - 1 - M + 1)/(M - 1)(N - 1) = 1/29
→ (N - M) / (M - 1)(N - 1) = 1/29
→ 29(N - M) = (M - 1)(N - 1)
→ 29N - 29M = (MN - M - N + 1)
adding (29M + N - MN) both sides,
→ 29N + N - 29M + 29M - MN = MN - MN - M + 29M - N + N + 1
→ 30N - MN = 28M + 1
→ N(30 - M) = (28M + 1)
→ N = (28M + 1)/(30 - M)
since both M and N are positive integers , value of M will be less than 30 . (1 to 29)
now, at M = 1 ,
→ N = (28*1 + 1)/(30 - 1) = 29 / 29 = 1
but N ≠ M ( given N > M) .
putting value of M from 2 to 28 , value of N will not be an integer .
when we put M = 29 ,
→ N = (28 * 29 + 1)/(30 - 29)
→ N = 813 .
therefore,
→ (M + N)/28
→ (813 + 29) / 28
→ 842 / 28
→ (421/14) (Ans.)
Question 2) :- Consider the sequence defined by ak = 1/(k²+ k) for k ≥ 1. Given that aM + a(M + 1) + _____ + a(N – 1) = 1/29 for positive integers M and N with M < N. Find (M+N)/28 ?
Solution :-
→ ak = 1/(k² + k)
→ ak = 1/k(k + 1)
→ ak = (1/k) - 1/(k + 1) [since , (k + 1 - k) /k(k + 1) = 1/k(k + 1)]
so,
→ a(M) + a(M + 1) + _______ a(N - 1) = 1/29
→ {(1/M - 1/(M + 1)} + {1/(M + 1) - 1/(M + 2)} + ________ {1/(N - 1) - 1/(N - 1 + 1)} = 1/29
→ 1/M - 1/N = 1/29
→ (N - M)/MN = 1/29
→ 29(N - M) = MN
→ 29N - 29M - MN = 0
adding (29)² both sides, we get,
→ 29N + 841 - 29M - MN = 841
→ 29(N + 29) - M(29 + N) = 841
→ 29(N + 29) - M(N + 29) = 841
→ (N + 29)(29 - M) = 841
now, given that, M < N and both M and N are positive integers .
→ (N + 29)(29 - M) = 841 * 1 or 1 * 841 or 29 * 29
if we take 29 * 29 , (29 - M) becomes zero , which is not correct , if we take (N + 29) = 1, N will be (-28) which is not a positive integer .
taking (N + 29) as 841 and (29 - M) as 1 ,
→ N + 29 = 841
→ N = 841 - 29 = 812
and,
→ 29 - M = 1
→ M = 29 - 1 = 28
therefore,
→ (M + N)/28
→ (812 + 28)/28
→ 840/28
→ 30 (Ans.)
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