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Find the sum of the geometric sequence below: 5−20+80−320+⋯+5⋅(−4)5-20+80-320+\cdots+5\cdot(-4)

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Full solution

Q. Find the sum of the geometric sequence below: 5−20+80−320+⋯+5⋅(−4)5-20+80-320+\cdots+5\cdot(-4)
  1. Identify Terms: Identify the first term aa and the common ratio rr of the geometric sequence.a=5a = 5, r=−4r = -4
  2. Determine Number of Terms: Determine the number of terms nn in the sequence.\newlineThe last term is 5×(−4)n−15 \times (-4)^{n-1}, but we don't know nn yet.
  3. Find nth Term: Find the nnth term by comparing it to the last term given in the sequence.5⋅(−4)(n−1)=5⋅(−4)?5 \cdot (-4)^{(n-1)} = 5 \cdot (-4)^{?}We need to find the exponent that makes this term equal to the last term in the sequence.
  4. Use Sum Formula: Use the formula for the sum of a finite geometric series: Sn=a(1−rn)/(1−r)S_n = a(1 - r^n) / (1 - r), where nn is the number of terms.\newlineWe still need to find nn before we can use this formula.
  5. Realize Mistake: Realize there's a mistake because we don't have the last term to find nn. We can't find the sum without knowing the number of terms.

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