Fichas
Series Basics
Fichas del conjunto (21)
- 1What is a sequence?A list of numbers which follow specific patterns
- 2What is a series?The sum of a sequence
- 3What is sigma notation?
- 4What equation gives us the sum of the natural numbers?1/2 n(n+1)
- 5What equation gives us the sum of the squares?1/6 n(n+1)(2n+1)
- 6What equation gives us the sum of the cubes?1/4 n^2 (n+1)^2
- 7What is the sigma notation for the sequence: 1/3 + 1/4 +1/5 ..... +1/8 ?The rule is 1/r, so the sigma notation is (8Σr=3)1/r
- 8What is the sigma notation for the sequence: 3/4 + 5/9 + 7/16 + 9/25 + 11/36 ?Rule = 2r-1/r^2 , starts at r=2, so the sigma notation is (6Σr=2)2r-1/r^2
- 9What is the sigma notation of the sequence: 2-4+8-10+32-64 ?As the signs are alternating starting from positive, we use (-1)^r+1. Now if all the signs were the same, the rule would be 2r. We combine these to get the rule, so the sigma notation is (6Σr=1)(-1^r+1)2r
- 10If we have the notation (a=5Σa=2)b(a+1) , what is the sum?We sub in the values of a from 0 to 5 to get b+2b+3b+4b+5b+6b = 21b
- 11If we have the notation (q=5Σq=2)pq^2, what is the sum?We sub in the values of q from 2 to 5 to get 4p+9p+16P+25p = 54p
- 12Simplify (100Σr=70)r so it can be calculated without a calculator(100Σr=1)r - (69Σr=1)r = 2635
(as 1/2(100)(100+1) - 1/2(69)(69+1) = 2635) - 13Simplify (10Σr=1)2r+5 to work out the sum(10Σr=1)2r + (10Σr=1)5 , which is the same as 2(10Σr=1)r + 5(10Σr=1)1. Then 2(1/2(10)(10+1) + 5x10 = 160
- 14Simplify (20Σr=1)r^3-4r^2-90 to work out the sum(20Σr=1)r^3 - 4(20Σr=1)r^2 - 9(20Σr=1)1 ;
1/4(20)^2 (20+1)^2 - 4(1/6(20)(20+1)(40+1)) -9x20 = 32440 - 15When simplifying something such as (xΣr=y)r y is greater than 1, how should we do it?(xΣr=1)r - (y-1Σr=1)r
The rule is the 2nd sigma notation has the top number r-1 . - 16(nΣr=1)1 = .....n
- 17Simplify: 2(2x-1)(2x+1)(2x+3) + 2(2x+3)Common factors are: 2(2x+3)
Whats left: [(2x+1)(2x-1)+1]
Combine : 2(2x+3)[(2x+1)(2x-1)+1]
Simplify futher: 2(2x+3)[4x^2 -1+1] ,
8x^2(2x+3) - 18What is (3nΣr=1)r^2 ?The rule is r^2, which has the equation 1/6n(n+1)(2n+1) , so just sub in 3n to get 1/6(3n)(3n+1)(6n+1) , which is the same as 1/2n(3n+1)(6n+1)
- 19Factorise (nΣr=1)4r^3-12r4(nΣr=1)r^3 - 12(nΣr=1)r ;
4(1/4n^2(n+1)^2) - 12(1/2n(n+1)) ;
n^2(n+1)^2 - 6n(n+1). Now simplify;
Common : n(n+1) , Whats left: n(n+1)-6
Combine: n(n+1)[(n(n+1)-6] ; n(n+1)(n^2+n-6) = n(n+1)(n+3)(n-2) - 20Remember that (2nΣr=1)r^3 is the same as what?1/4(2n)^2(2n+1)^2
- 21Remember that 4n^4-8n^3+n^2+3n=0 is the same as....4n(n-1)(n+1/2)(n-3/2)
(the factors are 0 1 -1/2 3/2 but remember the coefficient 4!!)