U39 Arithmetic and Geometric Sequences: Essential Formulas

This section covers the fundamental definitions, formulas, and properties of arithmetic and geometric sequences, which are core topics in the DSE Mathematics curriculum.

1 Arithmetic Sequence

Definition and General Term

An arithmetic sequence is a sequence where the difference between any two consecutive terms is constant. This constant is called the common difference, denoted by $d$. If the first term is $a$, then the $n$th term, $T(n)$, is given by:

$$ T(n) = a + (n-1)d $$
$a$$a+d$$a+2d$$a+3d$$d$$d$$d$

Sum of the First $n$ Terms

The sum of the first $n$ terms of an arithmetic sequence, denoted by $S(n)$, can be calculated using the first term $a$ and the last term $l = T(n)$.

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

2 Geometric Sequence

Definition and General Term

A geometric sequence is a sequence where the ratio between any two consecutive terms is constant. This constant is called the common ratio, denoted by $r$, where $r \neq 0$. If the first term is $a$, then the $n$th term is:

$$ T(n) = ar^{\,n-1} $$
$a$$ar$$ar^2$$ar^3$$\times r$$\times r$$\times r$

Sum of the First $n$ Terms

The sum of the first $n$ terms of a geometric sequence depends on the value of the common ratio $r$.

$$ S(n) = \frac{a(1 - r^{\,n})}{1 - r}, \quad \text{for } r \neq 1 $$

If $r = 1$, the sequence is constant and $S(n) = na$.

Sum to Infinity

For a geometric sequence with $|r| < 1$, the sum of all its terms (sum to infinity) converges to a finite value.

$$ S(\infty) = \frac{a}{1 - r}, \quad \text{for } |r| < 1 $$

3 Key Properties and Applications

Arithmetic Mean and Geometric Mean

For three consecutive terms in a sequence, specific relationships hold.

  • Arithmetic Sequence: If $x$, $y$, $z$ are consecutive terms, then $y$ is the arithmetic mean of $x$ and $z$: $2y = x + z$.
  • Geometric Sequence: If $x$, $y$, $z$ are consecutive terms (with $x, y, z \neq 0$), then $y$ is the geometric mean of $x$ and $z$: $y^2 = xz$.

Problem-Solving Tips

When solving DSE problems involving sequences:

  1. Identify the sequence type by checking for a common difference $d$ or a common ratio $r$.
  2. Write down all given information using the standard notation: $a$, $d$ or $r$, $n$, $T(n)$, $S(n)$.
  3. Form equations based on the formulas and solve for the unknowns. Systems of equations are common.
  4. For word problems, define the sequence clearly. For example, "the salary increases by a fixed amount each year" suggests an arithmetic sequence.
  5. Remember the condition for the sum to infinity of a geometric series: $|r| < 1$.

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