Summary (Grade 12 NSC Matric Mathematics): Revision Notes

Summary

This revision note covers the key concepts and formulas for sequences and series that you need to master for your NSC Mathematics exam.

Arithmetic sequences

An arithmetic sequence is a sequence where there is a constant difference between consecutive terms.

Key properties:

• Common difference (d): The fixed amount added to each term to get the next term
• Formula for common difference: $d = T_n - T_{n-1}$
• General form: $a, a + d, a + 2d, a + 3d, ...$
• General formula: $T_n = a + (n - 1)d$
• Graph: When plotted, the points form a straight line

The common difference can be positive (increasing sequence) or negative (decreasing sequence).

Quadratic sequences

A quadratic sequence is a sequence where the second differences between terms are constant.

Key properties:

• Common second difference: The difference between consecutive first differences is constant
• General formula: $T_n = an^2 + bn + c$ (where $a$, $b$, and $c$ are constants)
• Graph: When plotted, the points form a parabola

Geometric sequences

A geometric sequence is a sequence where there is a constant ratio between consecutive terms.

Key properties:

• Common ratio (r): The fixed number that each term is multiplied by to get the next term
• Formula for common ratio: $r = \frac{T_n}{T_{n-1}}$
• General form: $a, ar, ar^2, ar^3, ...$
• General formula: $T_n = ar^{n-1}$
• Graph: When plotted, the points form an exponential curve

The common ratio can be positive or negative, and greater than or less than 1.

Sigma notation

Sigma notation is a mathematical shorthand used to represent the sum of terms in a sequence.

The symbol $\Sigma$ (sigma) with limits indicates:

• $\sum_{k=1}^{n} T_k$ means "sum all terms $T_k$ from $k = 1$ to $k = n$"
• $k$ is the index variable
• The bottom number (1) is the starting value
• The top number ($n$) is the ending value

This notation provides a concise way to write long sums without having to list every term.

Series

A series is the sum of a specific number of terms in a sequence.

Arithmetic series

The sum of terms in an arithmetic sequence:

Two key formulas:

• When you know the first term ($a$) and last term ($l$): $S_n = \frac{n}{2}[a + l]$
• When you know the first term ($a$) and common difference ($d$): $S_n = \frac{n}{2}[2a + (n - 1)d]$

Geometric series

The sum of terms in a geometric sequence depends on the value of the common ratio:

Two formulas:

• When $r < 1$: $S_n = \frac{a(1 - r^n)}{1 - r}$
• When $r > 1$: $S_n = \frac{a(r^n - 1)}{r - 1}$

Sum to infinity

For a convergent geometric series where $-1 < r < 1$, the sum approaches a fixed value as the number of terms approaches infinity.

Key points:

• Only works when $-1 < r < 1$ (the series must converge)
• Formula: $S_\infty = \frac{a}{1 - r}$
• If $|r| \geq 1$, the series diverges and has no finite sum