Arithmetic and geometric sequences Knowledge-Based Study

Study notes

1. Definition and General Term

   An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This fixed difference is called the common difference, denoted by $d$.

   If the first term of the sequence is $a_1$, then consecutive terms are generated by repeatedly adding $d$:

   • $a_1$
   • $a_2 = a_1 + d$
   • $a_3 = a_1 + 2d$
   • $a_n = a_1 + (n - 1)d$

   The formula for the $n$-th term (the general term) is therefore:

   $$a_n = a_1 + (n - 1)d$$

   where $n$ is a positive integer representing the term number.

   Example

   Consider the sequence 5, 8, 11, 14, ...

   • First term: $a_1 = 5$
   • Common difference: $d = 8 - 5 = 3$
   • The 20th term is: $a_{20} = 5 + (20 - 1)(3) = 5 + 19(3) = 5 + 57 = 62$

   Sum of the First n Terms

   An arithmetic series is the sum of the terms of an arithmetic sequence. The sum of the first $n$ terms, denoted $S_n$, can be determined using either of two equivalent formulas:

   $$S_n = \frac{n}{2}(a_1 + a_n)$$

   Substituting $a_n = a_1 + (n - 1)d$ into this equation gives:

   $$S_n = \frac{n}{2}\left(2a_1 + (n - 1)d\right)$$

   Example

   Find the sum of the first 15 terms of the arithmetic sequence 4, 9, 14, 19, ...

   • Here, $a_1 = 4$, $d = 5$, and $n = 15$.
   • Using the second formula:

   $$S_{15} = \frac{15}{2}\left(2(4) + (15 - 1)(5)\right) = \frac{15}{2}(8 + 70) = \frac{15}{2}(78) = 15 \times 39 = 585$$

   Definition and General Term

   A geometric sequence is a sequence in which each term after the first is found by multiplying the preceding term by a non-zero constant called the common ratio, denoted by $r$.

   If the first term is $g_1$, the terms follow the pattern:

   • $g_1$
   • $g_2 = g_1 \cdot r$
   • $g_3 = g_1 \cdot r^2$
   • $g_n = g_1 \cdot r^{n-1}$

   The general formula for the $n$-th term is:

   $$g_n = g_1 \cdot r^{n-1}$$

   To find the common ratio between consecutive terms, compute $r = \frac{g_{k+1}}{g_k}$ for any valid index $k$.

   Example

   Consider the sequence 3, 6, 12, 24, ...

   • First term: $g_1 = 3$
   • Common ratio: $r = \frac{6}{3} = 2$
   • The 8th term is: $g_8 = 3 \cdot 2^{8 - 1} = 3 \cdot 2^7 = 3 \cdot 128 = 384$

   Finite and Infinite Sums

   A geometric series is the sum of terms in a geometric sequence.

   Finite Geometric Series

   The sum of the first $n$ terms of a geometric sequence with $r \neq 1$ is given by:

   $$S_n = \frac{g_1(1 - r^n)}{1 - r}$$

   Infinite Geometric Series

   When the absolute value of the common ratio satisfies $|r| < 1$, the terms approach zero as $n$ increases without bound. The infinite series converges to a finite sum:

   $$S = \frac{g_1}{1 - r}$$

   If $|r| \ge 1$ and $g_1 \neq 0$, the infinite geometric series diverges and has no finite sum.

   Example

   Find the sum of the infinite series $12 + 6 + 3 + 1.5 + \dots$

   • First term: $g_1 = 12$
   • Common ratio: $r = \frac{6}{12} = \frac{1}{2}$
   • Since $|r| < 1$, the sum is:

   $$S = \frac{12}{1 - \frac{1}{2}} = \frac{12}{0.5} = 24$$