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# slides05 - ∑ = = = = n i k k i m k k n m k k a a a 1 10...

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1 CMSC 250 Summations and Products

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2 CMSC 250 Sequences – 2,4,6,8,… for i 1 a i = 2 i an infinite sequence – for i 1 b i = ( - 1) i an infinite sequence – for 1 i 6 c i = i + 5 a finite sequence
3 CMSC 250 Finding an explicit formula Figure out the formula for this sequence: ,... 25 1 , 16 1 , 9 1 , 4 1 , 1 - -

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4 CMSC 250 A Familiar Sequence Put two points on a circle; draw a line between them. How many regions? Put three points on a circle; draw a line between all pairs of points. How many regions? Put four points on a circle; draw a line between all pairs of points. How many regions? What is the pattern?
5 CMSC 250 Another Familiar Sequence 2 ) 1 ( 1 2 : 0 3 + + - = + = k k b k a k k k

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6 CMSC 250 Sum of items specified Product of items specified 6 5 4 3 2 1 6 1 2 2 2 2 2 2 2 + + + + + = = k k = = 5 1 ) 5 2 )( 4 2 )( 3 2 )( 2 2 )( 1 2 ( 2 k k
7 CMSC 250 Variable ending point n as the index of the final term for n = 2 for n = 3 = + + n k k n k 0 1

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8 CMSC 250 Telescoping series = + + - + n k k k k k 1 ) 2 1 1 ( 1 1 + Π = i i n i
9 CMSC 250 Properties Merging and splitting = = = + = + n m k k k n m k k n m k k b a b a ) (

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Unformatted text preview: ∑ + = = = + = n i k k i m k k n m k k a a a 1 10 CMSC 250 Properties, con't. ● Distribution ∑ ∑ = = = n m k k n m k k ca a c 11 CMSC 250 Properties, con't. ● Distribution ∑ ∑ ∑ ∑ = = = = = m i j n j i n j j m i i b a b a 1 1 1 1 12 CMSC 250 Properties, con't. ● Interchanging order of summation ∑ ∑ ∑ ∑ = = = = = m i ij n j n j ij m i a a 1 1 1 1 13 CMSC 250 Nesting of sum/product notation ● Variations (same or different??): ∑ ∑ = = m i n j ij Y 1 2 1 ) ( ∑∑ = = m i n j ij Y 1 2 1 ) ( ∑∑ = = m i n j ij Y 1 1 2 14 CMSC 250 Factorial ● Product definition: n ! = ∏ i = n ( n-1 )( n-2) … 2 • 1 ● Recursive definition: 0! = 1 n ! = n ( n-1)! ii i = 1 n...
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slides05 - ∑ = = = = n i k k i m k k n m k k a a a 1 10...

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