A geometric sequence {a_n} is defined by \(\mathrm{a}_\mathrm{n} = 360(0.2)^{\mathrm{n}-1}\) for n geq 1. What is the value of a_1?

1. TRANSLATE the problem information

• Given information:
  • Geometric sequence formula: \(\mathrm{a_n = 360(0.2)^{n-1}}\)
  • Need to find: \(\mathrm{a_1}\) (the first term)

• This tells us we need to substitute \(\mathrm{n = 1}\) into the formula

2. TRANSLATE the substitution

• Replace n with 1 in the formula:
  \(\mathrm{a_1 = 360(0.2)^{1-1}}\)

3. SIMPLIFY the exponent

• Calculate the exponent first: \(\mathrm{1 - 1 = 0}\)
• So we have: \(\mathrm{a_1 = 360(0.2)^0}\)

4. SIMPLIFY using the zero exponent rule

• Any non-zero number to the power 0 equals 1
• Therefore: \(\mathrm{(0.2)^0 = 1}\)
• So: \(\mathrm{a_1 = 360(1) = 360}\)

Answer: D. 360

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students forget or misapply the zero exponent rule

Many students incorrectly think that \(\mathrm{(0.2)^0 = 0}\) or that \(\mathrm{(0.2)^0 = 0.2}\). This leads them to calculate either \(\mathrm{a_1 = 360(0) = 0}\) or \(\mathrm{a_1 = 360(0.2) = 72}\).

This may lead them to select Choice A (0) or Choice C (72)

The Bottom Line:

This problem tests whether students can correctly apply the fundamental exponent rule that any non-zero number raised to the power 0 equals 1. The substitution part is straightforward, but the zero exponent often trips students up.