Question

Summation Homework 1 Summation Homework From the following program fragment given below: 2-1 sumo for to n-1 do for je to do for k0 to j-1 do sum sum + 1 end-for end-for end-for write sum derive function sum(n) in a closed form such that: (a) Values of sum(n) for any given n are exactly the same as the values of variable sum in the program above. (b) All sums resulting from your analysis are derived based on what was explained in the class. (c) All steps of yourk work must be shown without any shortcuts. (d) The result obtained for function sum(n) must be verified by induction. (e) The result obtained for function sum(n) must be factored in terms of factors each in the form of (n-a), where a is a real constant.

Answer 1

i 2 times sumo for to n-1 for it to 12-1 do for k0 to j-1 do sum sum + 1 end-for end-for end-for write sum Value of sum voorable assigned o [sumzo] do → A loop that runs for a times for each value of į this runs for 2 for each value of j this runs for j liny time value of sum is sum is increased by L & each

امر د ۱- Sumon) = = (1) 3 33 : ؟ ۹۰ دو 2. 1- 1-1 نان ع ع ذ ما يتم !ته 0: ن ہ-1 (2-1) i2 2 =0 6 ۔ ) ۔ کر۔ ق ۔ م م ) . . ما به کره (1- ط) ۔ 10 () (-) (۲۰ " - - [20p-) + S( - - sn-1) - - 2] ۲-۱ (۴) (n-2) +) ۲-) : (1-2)n-1) (n)n+1) 2n- 10 h - 25 ( و

Summation Expansion Equivalent Value n k = 1+2+3+4+..+n = (ทำ + n) / 2 = (1/2) + (1/2)n k=1 = 1 + 4 +9+ 16 ++ n = (1/6)n(n+1)(2n+1) = (1/3)n + (1/2) + (1/6)n k=1 n ΣΕ = 1 +8 +27+ 64 +..+ n = (1/4) - (1/2)n + (1/4)n n Lะ = 1 + 16 + 81 + 256 +.+ = (1/5)n + (1/2) + (1/3)n ( - (1/30)n k=1

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