, find
.
I came across this sum in An Imaginary Tale by Nahin and I was fascinated.
Let and
.
Remember![]() Hence, ![]() ![]() ![]() Therefore, ![]() and ![]() ![]() ![]() |
Which means,
Let’s try a few partial sums
Hence,
What happens as ?
Because we know is undefined.
, find
.
I came across this sum in An Imaginary Tale by Nahin and I was fascinated.
Let and
.
Remember![]() Hence, ![]() ![]() ![]() Therefore, ![]() and ![]() ![]() ![]() |
Which means,
Let’s try a few partial sums
Hence,
What happens as ?
Because we know is undefined.
Filed under Identities, Interesting Mathematics, Puzzles, Sequences, Trigonometry
If
, then find
.
My first thought was to solve for , but it doesn’t factorise easily, and I didn’t want to find the fifth power of an expression involving surds
, there must be an easier way.
Because , we can divide by
Hence
(1)
What is the expansion of ?
Using the binomial expansion theorem
Therefore
(2)
Let’s do it again for
(3)
Substitute into
Remember
Therefore
This would be a good extension question for students learning the binomial expansion theorem. We also use this technique for trigonometric identities using complex numbers.
Filed under Algebra, Binomial Expansion Theorem, Puzzles
If
, what does
equal?
Two positive numbers are such that their difference, their sum, and their product are in the ratio
. What is the smaller of the two numbers?
Let and
be the two numbers. Then
(1)
(2)
(3)
Add equation and
together to eliminate the
(4)
From
, substitute for
into equation
.
(5)
Substitute into equation
.
Hence, or
.
When ,
and
Therefore the smaller number is .
Filed under Algebra, Puzzles, Ratio, Solving Equations