Question

Write the sum using summation notation, assuming the suggested pattern continues.

Answer 1

Answer:

$\sum_{n = 1} 2*(-5)^{n-1}$

Step-by-step explanation:

An arithmetic sequence is of the form:

$A_n = A_{n-1} + d$

While a geometric sequence is of the form:

$A_n = A_{n-1}*r$

notice that first, we have a change of sign in our sequence, so we already can discard the arithmetic sequence.

In fact, the pattern is kinda easy to see.

The first term is:

A₁ = 2

the second term is:

A₂ = -10

notice that:

A₂/A₍ = r = -10/2 = -5

The third term is:

A₃ = 50

the quotient between the third term and the second term is:

A₃/A₂ = 50/-10 = -5

Whit this we can already conclude that the n-th term of our sequence will be:

$A_n = A_{n-1}*(-5)$

Then the summation will be something like:

$\sum_{n = 1} A_n = A_1 + A_2 + A_3 + ... = 2 - 10 + 50 - ...$

We can write:

$A_n = A_{n-1}*(-5) = (A_{n-2}*(-5))*(-5)) = A_1*(-5)^{n-1} = 2*(-5)^{n-1}$

Then the summation is just:

$\sum_{n = 1} 2*(-5)^{n-1}$