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Arithmetic Series
STEP 1999, Math I, #1: How many integers greater than or equal to zero and less than a million are not divisible by 2 or 5?
What is the average value of these integers?
How may integers greater than or equal to zero and less than 4179 are not divisible by 3 or 7?
What is the average value of these integers?
Geometric Series
A geometric series is a series of the form
Each term is just
times the preceding term. The number is referred to as the common ratio. It can always be found by dividing
one term by the previous term:
Green Question: Suppose in a geometric series
and . Find .
Solution: It's easy just to count jumps from one term to the next. You need to multiply
by the common ratio three times to get to .
So
.
Then
We can find the sum of a finite geometric series as well since it's so predictable. Let
Then
We can subtract these two sums and most of the terms will cancel out:
so
Notice that the exponent on the
in the numerator is equal to the number of terms you are adding up.
For example, consider the series
Applying our sum formula we get
You can see that in the above series, the terms continually grow larger. That's because the common ratio is greater than 1. Therefore we
can only define an infinite geometric series if the the ratio is less than one.
where
In this case
So that we can calculate the sum as
Red Question: (P. Zeitz) Find (fairly) simple formulas for
and
STEP 2004, Math I, #7: (i) The function
is defined for by
where
, and for .
Simplify , and hence show that
Hence show that .
(ii) The function
is defined for by
where
, , , and for .
Obtain an expression for
as the sum of two algebraic fractions and determine in terms of .
STEP 2000, Math III, #8: The sequence
is defined by , , and
for
Prove by induction that
for
Hence show that
where
STEP 2004, Math III, #4: The triangle OAB is isosceles, with OA = OB and angle AOB =
where . The semi-circle
has its centre at the midpoint of the base AB of the triangle, and the sides OA and OB of the triangle are both tangent to the semi-circle.
, , , ... are circles such that is tangent to and to sides OA and OB of the triangle.
Let
be the radius of . Show that
Let
be the total area of the semi-circle and the circles , , , ... Show that
Show that there are values of
for which is more than four fifths of the area of triangle OAB.
STEP 2002, Math I, #8: I borrow
pounds at interest rate % per year. The interest is added
at the end of each year. Immediately after the interest is added, I make a repayment. The amount I
repay at the end of the
th year is pounds and the amount I owe at the beginning of the th year is
pounds (with ) .
Express in terms of ( = 1, 2, ... , ) , and and show that, if I pay off the loan in
years with repayments given by
,
where
is constant, then .
If instead I pay off the loan in
years with equal repayments of pounds, show that
,
and that in the case , .
STEP 1998, Math II, #5: Define the modulus of a complex number
and give the geometric interpretation of for two complex
numbers
and . On the basis of this interpretation establish the inequality
.
Use this result to prove, by induction, the corresponding inequality for
.
The complex numbers
satisfy . Prove that the equation
has no solution
with .
STEP 2011, Math II, #7: The two sequences
, , , . . . and , , , . . . have general terms
and ,
respectively, where
and .
(i) Show that
and give a corresponding result for
.
(ii) Show that, if
is odd,
and give a corresponding result when
is even.
(iii) Show that, if
is even,
and give a corresponding result when
is odd.
STEP 2008, Math III, 8: (i) The coefficients in the series
satisfy a recurrence relation of the form . Write down the value of .
By considering , find an expression for the sum to infinity of (assuming that it exists.) Find also an expression for the sum to
the first
terms of .
(ii) The coefficients in the series
. Find an expression for the sum to infinity of
satisfy a recurrence relation of the form
(assuming that it exists.)
By expressing
in partial fractions, or otherwise, find an expression for the sum to the first terms of .
STEP 1999, Math III, #1: Consider the cubic equation
where and .
(i) If the three roots can be written in the form , and for some constants and , show that one
root is
, show that
is a root and that the product of the other two roots is .
Deduce that the roots are in geometric progression.
(iii) Find necessary and sufficient conditions involving
and that .
(ii) If
, and for the roots to be in arithmetic progression.
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