Math, asked by yanarajan24, 1 month ago

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

Answered by SidMistry
3

Answer:

k=29

Step-by-step explanation:

Answered by RvChaudharY50
4

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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