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This projection combines two distinct pieces of compound-growth arithmetic. The first term, P times (1 + r) to the power n, is the future value of a single lump sum left untouched. The second, C times ((1 + r) to the power n minus 1) divided by r, is the future value of an ordinary annuity: a level contribution C made at the end of each year, where each deposit compounds for one fewer year than the one before it. That second expression is why the r appears in the denominator, and why the formula must be handled separately when r equals zero, in which case the balance is simply principal plus contributions times years. The rate you enter dominates everything else over long horizons. US large-cap equities have returned roughly 10 percent a year nominally over the long run, close to 7 percent after inflation, while investment-grade bonds have delivered something nearer 2 to 5 percent; a balanced 60/40 portfolio is usually modelled at 6 to 8 percent. The Rule of 72 gives a sanity check, since 72 divided by the return approximates the doubling time, about 10.3 years at 7 percent. The important caveat is that this model is deterministic. Real markets deliver a sequence of returns, not a constant one, and the order matters enormously once withdrawals begin. The formula also ignores taxes, fund expense ratios, advisory fees, and inflation, so enter a real return if you want the answer expressed in today's purchasing power.

Compound interest is far older than the notation used to express it. Babylonian tablets record interest accruing on loans, and medieval European merchants worked with doubling times long before the algebra was formalised. Luca Pacioli, the Franciscan friar whose Summa de Arithmetica appeared in Venice in 1494, is generally credited with the earliest known statement of the Rule of 72 as a mental shortcut for estimating how long money takes to double. The first systematic tables came from Richard Witt, whose Arithmeticall Questions was published in London in 1613 and computed compound interest and annuity values directly rather than treating them as a curiosity, making it a foundational text for actuarial practice. The annuity term in this formula became central to ordinary household finance only in the twentieth century, as regular payroll-based saving replaced lump-sum investing. Rigorous evidence about what return to assume arrived with Roger Ibbotson and Rex Sinquefield, whose 1976 studies in the Journal of Business assembled year-by-year returns on US stocks, bonds, bills, and inflation back to 1926 and gave planners a defensible historical baseline for the first time. The subsequent spread of index funds, following John Bogle's launch of the first retail index fund at Vanguard in 1976, made the modelled market return something an ordinary saver could actually capture rather than a theoretical benchmark.