Solve the following summation problems using the formulas provided: 1. 50 2. 5. 2iu4 (4i 3) 3. 36 4. 6. Eieo (j (-1)") 5. 7. 2je2(2' + 4) Formulas required to be used: TABLE Some Useful Summation Formulae. 1. Sum: Closed Form: ar^k (r ≠0) ar^(n+1) - 1 / (r - 1) (r ≠1) k=0 2. Sum: Closed Form: 2^(7k) (r ≠0) (i = 0) wr^(n+1) - 1 / (r - 1) (r ≠1) 3. Sum: Closed Form: c(n-i+1) 2^(u+2) 4 ? n(n + 1)(2n +1) 4. Sum: Closed Form: 6 6 k^3 26+42
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For problem 5, we have the summation: $$\sum_{i=0}^{50} 2i^4(4i^3)$$ Show more…
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(1) Calculate ∑i=1 to 2 ∑j=1 to 3 i^2j. (2) Calculate ∑i=5 to 11 i^2 using the table below: Table 2: Some Useful Summation Formulae: Sum Closed Form
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Problem 1 We mentioned in class the sum of the first n powers S(n, k) := 1^k + 2^k + ··· + n^k for k = 1, 2, 3, which is given by S(n, 1) = 1 + 2 + ··· + n = n(n + 1) / 2 , S(n, 2) = 1^2 + 2^2 + ··· + n^2 = n(n + 1)(2n + 1) / 6 , S(n, 3) = 1^3 + 2^3 + ··· + n^3 = ( n(n + 1) / 2 )^2 , Check the above formula explicitly for n ≤ 4. Find a formula for S(n, 4) and S(n, 5) and check your formula for n ≤ 4. Can you find a formula for arbitrary k?
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