4.5 Q-3

Exercise 4.5 — Question 3

This exercise covers harmonic sequences and series and the standard summation formulas for natural numbers, their squares, and cubes.

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Key Concepts

Harmonic Sequence

A sequence $H_1,H_2,H_3,\dots$ is called a harmonic sequence if the reciprocals $\frac{1}{H_1},\frac{1}{H_2},\frac{1}{H_3},\dots$ form an arithmetic sequence.

General Term: $H_n=\frac{1}{a+(n-1)d}$ where $a$ is the first term and $d$ is the common difference of the corresponding arithmetic sequence.

Example: $\frac{1}{2},\frac{1}{5},\frac{1}{8},\frac{1}{11},\dots$ is harmonic because $2,5,8,11,\dots$ is arithmetic with $a=2$, $d=3$.

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Standard Summation Formulas

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Worked Examples

Example 1 — Finding a Term of a Harmonic Sequence

Problem: Find the 10th term of the harmonic sequence $\frac{1}{3},\frac{1}{7},\frac{1}{11},\dots$

Solution:

• Corresponding AP: $3,7,11,\dots$ with $a=3$, $d=4$
• $a_{10}=3+(10-1)(4)=3+36=39$
• Therefore $H_{10}=\frac{1}{39}$

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Example 2 — Using Summation Formulas

Problem: Find $\sum _{k=1}^{10}{k}^2$.

Solution: ${\sum }_{k=1}^{10}{k}^2=\frac{10\cdot 11\cdot 21}{6}=\frac{2310}{6}=385$

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Example 3 — Sum of Cubes

Problem: Find $1^3+2^3+3^3+\cdots +8^3$.

Solution: ${\sum }_{k=1}^8{k}^3={\left[\frac{8\cdot 9}{2}\right]}^2=(36{)}^2=1296$

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Example 4 — Combined Summation

Problem: Evaluate $\sum _{k=1}^n(2k^2+3k+1)$.

Solution: $=2\sum k^2+3\sum k+\sum 1$ $=2\cdot \frac{n(n+1)(2n+1)}{6}+3\cdot \frac{n(n+1)}{2}+n$ $=\frac{n(n+1)(2n+1)}{3}+\frac{3n(n+1)}{2}+n$

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