Question

Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence whose nth term is an = (3)5n Find a2/a3, a1/a2, a4/a3 and a5/a4 What do you observe?

Accepted Answer

Identify the general term of the sequence given by an = 3 	imes 5^n. This means each term is 3 times 5 raised to the power of n. Write expressions for the specific terms needed: $$a_1 = 3 	$$imes $$5^1$$, $$a_2 = 3 	$$imes $$5^2$$, $$a_3 = 3 	$$imes $$5^3$$, $$a_4 = 3 	$$imes $$5^4$$, and $$a_5 = 3 	$$imes $$5^5. $$Calculate each ratio by dividing the corresponding terms: \frac{$$a_2$$}{$$a_3$$} = \frac{3 	imes $$5^2$$}{3 	imes $$5^3$$}, \frac{$$a_1$$}{$$a_2$$} = \frac{3 	imes $$5^1$$}{3 	imes $$5^2$$}, \frac{$$a_4$$}{$$a_3$$} = \frac{3 	imes $$5^4$$}{3 	imes $$5^3$$}, and \frac{$$a_5$$}{$$a_4$$} = \frac{3 	imes $$5^5$$}{3 	imes $$5^4$$}. Simplify each ratio by canceling the common factor 3 and applying the properties of exponents: \frac{5^m}{5^n} = $$5^{m-n}. $$For example, \frac{$$a_2$$}{$$a_3$$} = $$5^{2-3}$$ = $$5^{-1}. $$Observe the simplified ratios to identify any pattern or relationship, such as whether the ratios are constant or follow a specific rule related to the powers of 5.

Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence whose nth term is a_n = (3)5ⁿ Find a₂/a₃, a₁/a₂, a₄/a₃ and a₅/a₄ What do you observe?

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