Harmonic Sequence

Harmonic Sequence

and other sequences

Harmonic Sequence

A harmonic sequence is a sequence where in the reciprocal of its terms form an arithmetic sequence.

Multiple Choice

Which of the terms below may form a harmonic sequence?

• 3, 5, 7, 9, ....

• 1/2. 1/4, 1/8, 1/16, ...

• 1/3, 1/6, 1/9, 1/12, ....

Multiple Choice

Which of the statement is true?

• The reciprocals of the terms of an arithmetic sequence forms a harmonic sequence.

• The reciprocals of the terms of a harmonic sequence forms an arithmetic sequence.

Finding the nth term of a harmonic sequence

• Get the reciprocal of each term to form an arithmetic sequence.

• Now that you have an arithmetic sequence, use the formula

   $a_n=a_1\ +\left(n-1\right)d$  

• Once you have the nth term of the arithmetic sequence, get its reciprocal. 

• The reciprocal the nth term of the arithmetic sequence is the nth term of the harmonic sequence.

Fill in the Blank

Find the 10th term in the harmonic sequence that starts with

 $4,\ 2,\ \frac{4}{3},\ 1,\ ...$  

Fill in the Blank

Find the 13th term of the harmonic sequence that starts with 1, 3/5, 3/7, 1/3.

Harmonic mean

term or terms in between two nonconsecutive terms in a harmonic sequence

Harmonic Mean

• Similar to finding the nth term of a harmonic sequence, you start by getting the reciprocal of each term to form an arithmetic sequence.

• After that, follow the steps in finding the arithmetic mean/s.

• To get the final answer/s, find the reciprocal of the arithmetic mean/s.

Fill in the Blank

Find the harmonic mean of 5 and 10.

Fill in the Blank

Find the harmonic mean of -4 and 2.

Other Sequences

• Fibonacci Sequence

• Triangular Number Sequence

• Square Number Sequence

• Cube Number Sequence

Fibonacci Sequence

1, 1, 2, 3, 5, 8, ...

Open Ended

Let's have a short quick look into the Fibonacci sequence. You have 15 minutes to look into the life of Fibonacci and the importance or the presence of the Fibonacci sequence in our life. Share what you have found here.

Triangular Number Sequence

1, 3, 6, 10, 15, ...

Try to figure out the formula to find the nth triangular number.

Square Numbers Sequence

1, 4, 9, 16, 25, ...

What is the formula to find the nth square?

Cube Numbers Sequence