Question

Rezolvaţi în N ecuația : 1/3+1/15+1/35+...+1/(2n-1)x(2n+1)=12/25​

Answer 1

Răspuns:

n = 12

Explicație pas cu pas:

folosim formula:

$\frac{1}{2n - 1} - \frac{1}{2n + 1} = \frac{2n + 1 - (2n - 1)}{(2n - 1)(2n + 1)} = \frac{2}{(2n - 1)(2n + 1)}$

$\frac{1}{2n - 1} - \frac{1}{2n + 1} = 2 \cdot \frac{1}{(2n - 1)(2n + 1)}$

$\boxed { \red{\frac{1}{(2n - 1)(2n + 1)} = \frac{1}{2} \cdot \Big(\frac{1}{2n - 1} - \frac{1}{2n + 1}\Big)}}$

→

$\frac{1}{3} + \frac{1}{15} + \frac{1}{35} + ... + \frac{1}{(2n - 1)(2n + 1)} = \frac{12}{25}$

$\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + ... + \frac{1}{(2n - 1)(2n + 1)} = \frac{12}{25}$

$\frac{1}{2} \cdot \Big(\frac{1}{1} - \frac{1}{3}\Big) + \frac{1}{2} \cdot \Big(\frac{1}{3} - \frac{1}{5}\Big) + \frac{1}{2} \cdot \Big(\frac{1}{5} - \frac{1}{7}\Big) + ... + \frac{1}{2} \cdot \Big(\frac{1}{2n - 1} - \frac{1}{2n + 1}\Big) = \frac{12}{25}$

$\frac{1}{2} \cdot \Big(\frac{1}{1} - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + ... + \frac{1}{2n - 1} - \frac{1}{2n + 1}\Big) = \frac{12}{25}$

$\frac{1}{1} - \frac{1}{2n + 1} = \frac{24}{25} \iff \frac{2n + 1 - 1}{2n + 1} = \frac{24}{25}$

$\frac{2n}{2n + 1} = \frac{24}{24 + 1} \iff 2n = 24$

$\implies \red{\bf n = 12}$