Arithmetic Sequence

The Proof

• Sn = n/2 [2a+(n-1)d]
• Sn = (a) + (a+d) + (a+2d) + . . . + [a+(n-2)d] + [a+(n-1)d] (1)
• Sn = [a+(n-1)d] + [a+(n-2)d] +. . . + [a+d] + [a] (2)
• (1) + (2) gives:
• 2Sn = [2a+(n-1)d] + [2a+(n-1)d] + [2a+(n-1)d] + . . . + [2a+(n-1)d]
• 2sn = n[2a+(n-1)d]
• Sn = n/2 [2a+(n-1)d]

Properties

• Increase / decrease by same amount each term
• a - first term
• d - difference in terms
• n - number of terms
• E - sum of everything
• S - sum of terms
• Sum of sequence > n(n+1)/2
• The nth term > a+(n-1)xd

First term/Common difference

• Given sum of two set of terms
• 3 terms:
• (1) a+(term)d + a+(term)d + a+(term)d = given value
• (2) a+(term)d + a+(term)d + a+(term)d = other given value
• = a+(term)d = value 1 } Solve the two
• a+(term)d = value 2 } equations simultaneously
• Giving you a - first term
• d - difference in terms