Question

Write the sum using summation notation, assuming the suggested pattern continues.. . 5 - 15 + 45 - 135 + ...

Answer 1

Answer:

$\sum \limits_{n=1}^{\infty}(-3)^{n-1}\cdot 5$

Step-by-step explanation:

You are given the sum

$5-15+45-135+...$

Here,

$a_1=5\\ \\a_2=-15\\ \\a_3=45\\ \\a_4=-135\\ \\...$

Note that

$a_2=-3a_1\\ \\a_3=-3a_2\\ \\a_4=-3a_3\\ \\...$

Thus,

$a_n=-3a_{n-1}=(-3)(-3)a_{n-2}=... =(-3)^{n-1}a_1$

The sum using summation notation, assuming the suggested pattern continues is

$\sum \limits_{n=1}^{\infty}(-3)^{n-1}\cdot a_1=\sum \limits_{n=1}^{\infty}(-3)^{n-1}\cdot 5$