In a geometric sequence a_2=-144 and a_5=486 write the explicit formula for this sequence

Answer 1

Let's first find the common ratio, denoted by r.

We know that a_2 = a_1 * r = -144, and a_5 = a_1 * r^4 = 486.

Dividing the equation a_5 = a_1 * r^4 = 486 by the equation a_2 = a_1 * r = -144, we have (a_1 * r^4) / (a_1 * r) = 486 / -144.

Canceling out a_1, we have r^3 = (-486) / 144.

Simplifying the right side, we have r^3 = -3.375.

Taking the cube root of both sides, we have r = -1.5.

Now that we know the common ratio, we can write the explicit formula for this geometric sequence.

The general form of the explicit formula for a geometric sequence is a_n = a_1 * r^(n-1), where a_n is the nth term of the sequence, a_1 is the first term, r is the common ratio, and n is the position of the term.

Substituting our values into the formula, we have a_n = a_1 * (-1.5)^(n-1).

However, we still need to find the value of a_1 in order to have a complete explicit formula.

To find a_1, we need to substitute either a_2 or a_5 into the formula a_n = a_1 * (-1.5)^(n-1).

Let's use a_2 = -144. So, -144 = a_1 * (-1.5)^(2-1) = a_1 * (-1.5)^1 = -1.5a_1.

Dividing both sides by -1.5, we have a_1 = 96.

Now we can write the complete explicit formula for the sequence, which is a_n = 96 * (-1.5)^(n-1).