Problem

Find the account balance immediately after the $\text{sixth}$ $\$100$ month-end deposit at monthly growth factor $1.01$. State the growth exponents, included deposits, geometric series and formula, and approximate balance.

Neutral cash-flow timeline showing six equal positive $100 end-of-month deposits, valued immediately after deposit 6; growth exponents, series formula, and balance are withheld.

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What this problem is really about

Financial amounts can be added only after they are moved to the same valuation date. For equal end-of-period deposits, the earliest deposit has the most growth periods, while the deposit made at the valuation instant is included with exponent zero. Those descending exponents form a finite geometric series with one term for every deposit.

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Growth exponents: 5 for every deposit. Deposits included: all 6. Series/formula: 6·100(1.01)⁵. Balance: about $630.61.

Growth exponents: 6 through 1. Deposits included: all 6, each one month too long. Series/formula: 100·1.01[(1.01)⁵+…+1]. Balance: about $621.35.

Growth exponents: 5 through 1. Deposits included: first 5; final deposit omitted. Series/formula: 100[(1.01)⁵+…+1.01]. Balance: about $515.20.

Growth exponents: 5 through 0. Deposits included: all 6. Series/formula: 100[(1.01)⁵+…+1]=100((1.01)⁶-1)/0.01. Balance: about $615.20.

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Find the account balance immediately after the \(\text{sixth}\) \(\$100\) month-end deposit at monthly growth factor \(1.01\). State the growth exponents, included deposits, geometric series and formula, and approximate balance.

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