Математика22 мин

Sequences

Arithmetic and geometric sequences — general term, sum formulas, and how to spot the pattern.

Arithmetic sequences

a_n = a_1 + (n-1)d

S_n = \dfrac{n}{2}(a_1 + a_n) = \dfrac{n}{2}\bigl(2a_1 + (n-1)d\bigr)

Find a term

Sequence: 4, 7, 10, … Find the 20th term.

Жауап: 61

a_n = a_1 \cdot r^{\,n-1}

S_n = a_1 \cdot \dfrac{1 - r^n}{1 - r}\quad (r \neq 1)

S_\infty = \dfrac{a_1}{1 - r}\quad (|r| < 1)

Sum to infinity, only when

|r| < 1

Жаттығу

Find the sum to infinity of $1 + 1/2 + 1/4 + \dots$

Жауап пен түсініктемені көру

Жауап: 2

$a_1 = 1$ , $r = 1/2$ . $S_\infty = 1/(1-1/2) = 2$ .