ssrs barcode font download Obtain a general formula for the sum of squares of the consecutive integers 1 through n. in Software

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2.52 Obtain a general formula for the sum of squares of the consecutive integers 1 through n.
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Sum[k^2, {k, 1, n}] 1 (n)(1+ n)(1+2 n) 6
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Mathematica has memorized these standard formulas.
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2.53 Compute the product of the first 20 Fibonacci numbers.
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Fibonacci[i]
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Product[Fibonacci[i], {i, 1, 20}]
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9 692 987 370 815 489 224 102 512 784 450 560 000
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2.54 Compute the product of the natural logarithms of the integers 2 through 20. Obtain an approximation to 20 significant digits.
SOLUTION
N[Product[Log[i], {i, 2, 20}], 20] 1.3632878207490815857 106 1 2.55 Compute the sum 1 + 1 + 1 + 1 + 1 + 1 + . . . + 1 + 1 + 1 + . . . + 20 2 2 3 2 3
SOLUTION
( )(
Sum[1/j, {i, 1, 20}, {j, 1, i}] 41 054 655 739 024
1 j
i=1 j=1
2.56 Compute a numerical approximation of 1 + 1 1 + 1 + 1 . . . 1 + 1 + 1 + . . . + 1
2 3 2 3
SOLUTION
( 2 )(
NProduct[Sum[1/j, {j, 1, i}], {i, 2, 10}] 1871.44
1 j
i=2 j=1
2.8 Loops
Often you may need to repeat an operation or sequence of operations several times. Although Mathematica offers the ability to compute sums and products conveniently using the Sum and Product commands, there are times when your work may require the use of looping techniques. Mathematica offers three basic looping functions: Do, While, and For.
Do[expression,{k}] evaluates expression precisely k times. Do[expression,{i, imax}] evaluates expression imax times with the value of i changing from 1 to imax in increments of 1. Do[expression,{i, imin, imax}] evaluates expression with the value of i changing from imin to imax in increments of 1. Do[expression,{i, imin, imax, increment}] evaluates expression with the value of i changing from imin to imax in increments of increment. Do[expression,{i, imin, imax}, {j, jmin, jmax}] evaluates expression with the value of i changing from imin to imax and j changing from jmin to jmax in increments of 1. The variable i changes by 1 for each cycle of j. This is known as a nested Do loop. Do[expression, {i, imin, imax, i_increment}, {j, jmin, jmax, j_increment},...] forms a nested Do loop allowing for incrimination values other than 1. The last two forms of the command may be extended to three or more variables.
EXAMPLE 53
Do[Print["This line will be repeated 5 times."], {5}] This line will be repeated 5 times. This line will be repeated 5 times. This line will be repeated 5 times. This line will be repeated 5 times. This line will be repeated 5 times.
EXAMPLE 54
This example computes the sum of consecutive odd integers from 5 to 25. (Of course, the Sum command is more convenient.) mysum = 0; Do[mysum = mysum + k, {k, 5, 25,2}] mysum 165
Initialization of mysum. This step is important. It is not needed if the command Sum is used.
Basic Concepts
EXAMPLE 55 This example computes the sum of all fractions whose numerators and denominators are positive integers not exceeding 5.
fracsum = 0; Do[fracsum = fracsum + i/j, {i,1,5},{j,1,5}] fracsum 137 4
While[condition, expression] evaluates condition, then expression, repetitively, until condition is False.
If expression consists of multiple statements, they are separated by semicolons.
EXAMPLE 56
n = 1; While[n < 6, Print[n]; n ++] 1 2 3 4 5
n = n+1 may be used in place of n++ See page 37.
For[initialization, test, increment, expression] executes initialization, then repeatedly evaluates expression, increment, and test until test becomes False.
After initialization, the order of evaluation is test, expression, and then increment. The For loop terminates as soon as test gives False. If initialization, test, increment, or expression consists of multiple statements, they are separated by semicolons.
EXAMPLE 57
For[i = 1, i 5, i ++, Print[i]] 1 2 3 4 5
Although it is not a loop, the If instruction is often used in conjunction with other loop commands.
If[condition, true, false] evaluates condition and executes true if condition is True and executes false if condition is False. If[condition, true] evaluates condition and executes true if condition is True. If condition is False no action is taken and Null is returned. If[condition,, false] evaluates condition and executes false if condition is False. If condition is True no action is taken and Null is returned. (Note the double comma.) If[condition, true, false, neither] evaluates condition and executes true if condition is True, executes false if condition is False, and executes neither if condition is neither True nor False.
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