40 1. INTEGRATION, AREA, AND INITIAL VALUE PROBLEMS
e symbol
X
means “add up the things following the symbol.” So
5
X
iD1
i
is a symbolic way of saying “add up the numbers i from i D 1 to i D 5. e name of the symbol
is uppercase sigma – not to be confused with which is called lowercase sigma. Applying
P
we get
5
X
iD1
i D 1 C 2 C 3 C 4 C 5 D 15
which isn’t too bad. e problem is when we get something like
200
X
iD1
i D 1 C 2 C C199 C 200 D 20; 100
e incomparable German mathematician Carl Friedrich Gauss found the following shortcut:
n
X
iD1
i D
1
2
n.n C1/
How do you prove a formula like that is correct?
e usual technique is called mathematical induction.
Knowledge Box 1.14
Mathematical induction
Suppose we wish to prove a proposition P .n/ is correct for all n c.
en the following steps suffice
• Verify that P .c/ is true.
• Assume that, for some n, P .n/ is true.
• Show that, if P .n/ is true, then so is P .n C 1/