Grade 11 Math 1 Ch 4 (Sequences and Series) Notes

Chapter 4: Sequences and Series

This chapter introduces sequences and series, arithmetic progressions (A.P.), geometric progressions (G.P.) and harmonic progressions. It also explains formulas for finding general terms and sums of these progressions.

General Term:

$$u_n = a + (n-1)d$$

Where:

• \(a\) = first term
• \(d\) = common difference
• \(n\) = number of terms

Example:

Find the 10th term of:

\(2, 5, 8, 11, \dots\)

Here:

$$a = 2, \quad d = 3$$ $$u_{10} = 2 + (10 - 1)3 = 29$$

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Arithmetic Mean (A.M.)

If three numbers \(a, A, b\) are in A.P., then \(A\) is called the arithmetic mean.

Formula:

$$A = \frac{a + b}{2}$$

Example:

Arithmetic mean between 4 and 16:

$$A = \frac{4 + 16}{2} = 10$$

Inserting n Arithmetic Means

Between numbers \(a\) and \(b\):

Number of terms = \(n + 2\)

Common Difference:

$$d = \frac{b - a}{n + 1}$$

These terms form an A.P.

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4.3 Arithmetic Series

An arithmetic series is the sum of terms of an arithmetic progression.

Example:

$$1 + 3 + 5 + 7 + 9$$

Sum of First n Terms

\(S_n\)

• First term = \(a\)
• Common difference = \(d\)

Then:

$$S_n = \frac{n}{2}[2a + (n-1)d]$$

Alternative form:

Where \(l\) is the last term:

$$S_n = \frac{n}{2}(a + l)$$

Example:

Find the 6th term of:

\(2, 6, 18, 54, \dots\)

$$U_6 = 2(3)^5 = 486$$

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Geometric Mean (G.M.)

If \((a, G, b)\) are in G.P., then \((G)\) is called the geometric mean.

Formula:

$$G = \sqrt{ab}$$

Example:

Between 4 and 36:

$$G = \sqrt{4 \times 36}$$ $$G = 12$$

Inserting n Geometric Means

Between \((a)\) and \((b)\):

$$b = ar^{n+1}$$

Therefore:

$$r = \sqrt[n+1]{\frac{b}{a}}$$

Then construct the G.P.

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4.5 Geometric Series

A geometric series is the sum of terms of a geometric progression.

Example:

$$2 + 6 + 18 + 54 + \cdots$$

Sum of First n Terms

For \(r > 1\):

$$S_n = \frac{a(r^n - 1)}{r - 1}$$

Example:

Find the sum of first 5 terms:

\(2 + 6 + 18 + 54 = 162\)

$$S_5 = \frac{2(3^5 - 1)}{3 - 1}$$ $$S_5 = 242$$

Infinite Geometric Series Example:

$$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$$ $$S_{\infty} = \frac{a}{1 - r} = \frac{1}{1 - \frac{1}{2}} = 2$$