Problem

Decode and evaluate \(\sum_{k=0}^{5}\) \(3\cdot~2^k\). State the term count, first term, ratio, last term, finite-series formula, and sum.

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What this problem is really about

Decode sigma notation before applying a sum formula. Evaluate the summand at the lower bound for the first term, increase the index once to see the ratio, and count integer indices inclusively from lower to upper bound. Evaluating the final indexed term provides a quick check that the term count and exponent range agree.

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Term count: 6. Lower exponent and first term: 0 and 3. Ratio: 3. Last term: 3·3⁵=729. Formula: 3(1-3⁶)/(1-3). Sum: 1092.

Term count: 5. Lower exponent and first term: 1 and 6. Ratio: 2. Last term: 6·2⁴=96. Formula: 6(1-2⁵)/(1-2). Sum: 186.

Term count: 6. Lower exponent and first term: 0 and 3. Ratio: 2. Last term: 3·2⁵=96. Formula: 3(1-2⁶)/(1-2). Sum: 189.

Term count: 5. Lower exponent and first term: 0 and 3. Ratio: 2. Last term: 3·2⁴=48. Formula: 3(1-2⁵)/(1-2). Sum: 93.

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Decode and evaluate \(\sum_{k=0}^{5}\) \(3\cdot~2^k\). State the term count, first term, ratio, last term, finite-series formula, and sum.

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