function interest($initial=1000,$apr=0.015,$years=10) {
  $rate = $apr/12;
  $periods = 12*$years;
  $sum = $initial;
  for($i=0; $i<$periods;$i++) {
    $sum = $sum + $sum * $rate;
  }
  return $sum;
}

function interest($initial=1000,$apr=0.015,$years=10) {
  $rate = $apr/12;
  $periods = 12*$years;
  return $initial * pow(1 + $rate, $periods);
}

function interest($initial=1000,$addition=100,$apr=0.015,$years=10) {
  $rate = $apr/12;
  $periods = 12*$years;
  $sum = $initial;
  for($i=0; $i<$periods; $i++) { 
    // Add interest earned in previous term.
    // Add $addition in preparation for next compounding period.
    $sum = $sum + $sum*$rate + $addition;
  }
  // Remove last $addition because there is no addition at the end-of-term.
  return $sum - $addition;
}

/**
 * Compute the balance on an account after some period of time.
 * The account may receive an initial deposit and/or regular deposits
 * at the start of every compounding period, after the first.
 * The APR, duration, and number of compounding periods may be specified.
 *
 * @see http://en.wikipedia.org/wiki/Geometric_series
 * @param float $inital Initial deposit amount
 * @param float $addition Amount added at start of each compounding period
 * @param float $apr Annual percentage rate (percent/100) earned on balance
 * @param float $years Number of years ($years>0) to compute interest for.
 * @param uint $ppyr Number of compounding periods per year.
 * @return float Balance after $years time accruing interest at $apr rate.
 */
function interest($initial=1000,$addition=100,$apr=0.015,$years=10,$ppyr=12) {
  // Use geometric series to calculate the cumulative rate of return
  // with recurring, inter-period deposits (periods 0..N-1)
  // $initial is initial deposit, interest accrues immediately.
  // $rate is interest earned per period (percent/100).
  // $addition is amount added at beginning of period after first period.
 
  // Over time, interest accrues on the additional deposits:
  // let R = (1 + rate)
  // v0 = 0  -- we'll deal with the initial deposit elsewhere.
  // v1 = v0 * R + additional
  // v2 = (v1) * R + additional
  // v2 = (v0 * R + add) * R + add
  // v2 = 0 + add*R + add
  // v2 = add*R + add
  // v3 = v2 * R + add
  // v3 = (add*R + add)*R + add
  // v3 = add(R^2 + R + 1)
  // v4 = v3 * R + add
  // v4 = (add*R*R + add*R + add)*R + add
  // v4 = add(R^3+R^2+R^1+R^0)
  // v5 = v4 * R + add
  // v5 = (v3 * R + add) * R + add
  // v5 = ((v2 * R + add) * R + add) * R + add
  // v5 = (((v1 * R + add) * R + add) * R + add) * R + add
  // v5 = ((((v0 * R + add) * R + add) * R + add) * R + add) * R + add
  // v5 = add * (R^4 + R^3 + R^2 + R^1 + R^0)
  // ... using geometric series ... a = add, r = (1+rate)
  // vN = a * (1 - pow(r,N)) / (1-r)
 
  $rate = $apr / $ppyr;
  $periods = $years * $ppyr;
 
  // Geometric series:
  // a + ar + ar^2 + ar^3 + ... + ar^n 
  //   = sum(k=0..n)[ar^k]
  //   = a * (1-r^(n+1)) / (1-r)
  // Where:
  //   a = $addition
  //   r = 1 + $rate
  //   n = $periods - 1
  //   k = [0..n]
  $a = $addition;
  $r = 1 + $rate;
  // Compute series from first period to last period
  $geom = $a * (1 - pow($r,$periods)) / (1-$r);
  // Remove $add from last period because next period didn't begin yet.
  $additional = $geom - $addition;
 
  // Compute interest on initial deposit + additional interest from above.
  return $initial * pow(1+$rate,$periods) + $additional;
}

function interest($initial=1000,$addition=100,$apr=0.015,$years=10,$ppyr=12) {
  $R = 1 + $apr / $ppyr;
  $pow = pow( $R, $years * $ppyr );
  return ($initial * $pow) + ($addition * (1 - $pow) / (1-$R) - $addition);
}