Sum of n numbers: 1+2+3+ ... + n = n(n+1) / 2
sigma sigma j
= sigma n(n+1) / 2
= n*n(n+1) / 2
= O(n3)
Give a closed form for the following double sum: Justify your answer.
Find a closed form for sn=∑i=1n(5i−1). What is the first term of this sum? What is the nth term of the sum? If we write the sum both forwards and backwards and pair the terms vertically, what is the identical sum for each pair? Finally, give the closed form for sn=∑i=1n(5i−1). Find a closed form for sn=∑i=1n(8i−9). sn= Find a closed form for sn=∑i=1n(17−9i). sn= Find a closed form for sn=∑i=1n(−4−3i). sn=
7. (0.2 pts) Answer True/False and justify your answer. a) b) (0.1 pts) Double-stranded RNA preferentially form B-form polymers (0.1 pts) The dipoles of water molecules decrease the strength of other electrostatic interactions
(1 point) Express the following sum in closed form /2 Σ (342k) k-l Hint: Start by multiplying out (32k)2. Note: Your answer should be in terms of n.
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5. Answer the following questions. Justify your answers. A. (8pts) If the nth partial sum of a series an is equal to sn = (1+0)") what is the sum of the series? en B. (8pts) Determine if the series 2n=1zten is convergent or divergent. C. (4pts) Can you make an infinite series of nonzero terms that converges to any number you want? Explain.
2. Find the sum of the serie 3-5*+1-6-24+2 -. Give your answer in exact form. 74
1- What is a geometric progression? Give an example to justify your answer. 2- What is an arithmetic progression? Give an example to justify your answer. 3- What is a recurrence relation. 4- What isthe method that we might use to solve recurrence relations ? 5- What is the difference between a geometric progression and geometric serie. Justify your answer.
1) In Young's double-slit experiment what is the condition for a bright fringe? Give your answer both in words and in equation form. 2) is bright fringe caused by constructive or destructive interference between the waves from two slits? 3) what is the condition for destructive interference in a single slit diffraction pattern? Give your answer both in words and in equation form.
4. (5 points) Is the following grammar ambiguous? Justify your answer (give a string and derive it with two leftmost derivations using G). If it is ambiguous, rewrite this grammar to an unambiguous one (hint: recall "dangling else" as we discussed in the class)
2. Use the summation facts and the known closed forms to find a closed form for the following sum: 1+ 4 + 8 + 14 + ... + (2n+2") Simplify the expression Final Answer:
Determine if the given set is a subspace of P4. Justify your answer. All polynomials of degree at most 4, with integers as coefficients. Complete each statement below. The zero vector of P4 in the set because zero an integer The set v closed under vector addition because the sum of two integers an integer The set closed under multiplication by scalars because the product of a scalar and an integer an integer Is the set a subspace of P4?...