Math Problem Statement

A sequence is defined recursively by the given formulas. Find the first five terms of the sequence: a_n = a_(n-1)/5 and a_1 = -20.

Solution

The given problem describes a sequence defined recursively with the formulas:

an=an15,a1=20.a_n = \frac{a_{n-1}}{5}, \quad a_1 = -20.

We are tasked to find the first five terms of the sequence. Let's compute them step by step:

  1. a1=20a_1 = -20 (given).

  2. a2=a15=205=4a_2 = \frac{a_1}{5} = \frac{-20}{5} = -4.

  3. a3=a25=45=0.8a_3 = \frac{a_2}{5} = \frac{-4}{5} = -0.8.

  4. a4=a35=0.85=0.16a_4 = \frac{a_3}{5} = \frac{-0.8}{5} = -0.16.

  5. a5=a45=0.165=0.032a_5 = \frac{a_4}{5} = \frac{-0.16}{5} = -0.032.

Final Result:

a_1 &= -20, \\ a_2 &= -4, \\ a_3 &= -0.8, \\ a_4 &= -0.16, \\ a_5 &= -0.032. \end{aligned}$$ Would you like a detailed explanation of the steps? Let me know if you have questions! --- ### Related Questions: 1. What is the general form for this sequence if we need a formula for $$a_n$$? 2. How can we verify convergence of this sequence? 3. What is the sum of the first $$n$$ terms of this sequence? 4. How does the value of $$a_1$$ influence the behavior of the sequence? 5. Can this sequence be represented as a geometric series? **Tip:** Always check recursive steps carefully, as even small errors can propagate through calculations!

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Math Problem Analysis

Mathematical Concepts

Sequences and Series
Recursive Formulas
Arithmetic Operations

Formulas

a_n = a_(n-1)/5

Theorems

-

Suitable Grade Level

Grades 9-11