Question

what are the explicit equation and domain for a geometric sequence with a first term of 5 and a second term of -10

Answer 1

Answer:

• The explicit equation is given by:

         $a_n=5(-2)^{n-1}$

• The domain of the geometric sequence is:  All the natural numbers (i.e. n≥1)

Step-by-step explanation:

Explicit formula--

The explicit formula is a formula which is used to represent the nth term of a sequence in terms of the variable n.

It is given that:

The first term of the sequence is 5 and the second term is -10.

This means that if a denotes the first term and r denotes the common ratio.

The geometric sequence is given by: a,ar,ar²,ar³,....

i.e. the nth term of the sequence is given by:

$a_n=ar^{n-1}$

Then we have:

$a=5$

and

$ar=-10\\\\i.e.\\\\5\times r=-10\\\\i.e.\\\\r=\dfrac{-10}{5}\\\\i.e.\\\\r=-2$

Hence, the nth term of the sequence is given by:

$a_n=5(-2)^{n-1}$

We know that the domain of a geometric sequence is the set of all the natural numbers.

( since the term a_n is defined for all the natural numbers ).

Answer 2

$a_1=5;\ a_2=-10;\ r=a_2:a_2\to r=-10:5=-2\\\\a_n=a_1r^{n-1}\\\\\boxed{a_n=5\cdot(-2)^{n-1}}$