yr2 chap 3 sequences + series

• formula for an arithmetic sequence:

  Un = a + (n-1)d
• formulae for sum of an arithmetic sequence:

  Sn = n/2 (2a + (n-1)d)

  Sn = n/2 (a+l)
• formula for a geometric sequence:

  Un = ar^(n-1)
• formula for sum of a geometric sequence:

  Sn = a(1-r^n)/1-r
• the three types of sequence are arithmetic, geometric and iterative
• formulae for an iterative sequence:

  U(n) = U(n-1) + d

  U(n) = rU(n-1)
• in sigma notation, the number on the top represents the number of terms
• in sigma notation, the number on the bottom with r = ... represents the first number that you substitute in for the first term
• in sigma notation, the part after the sigma is the nth term of the sequence, it is what you substitute the numbers into to get the terms
• to prove the formula Sn = n/2 (2a + (n-1)d):

  • write out the first two and last two terms
  • write out the sum using + ... + to show the other terms, label this equation 1
  • write out the sum again backwards using + ... + to show the other terms, label this equation 2
  • add equation 1 and equation 2 together to get 2Sn, show 2 identical terms then + ... + then 2 identical terms
  • collect the identical terms together using a multiple of n
  • divide both sides by 2 to get Sn