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Find the sum of the multiples of 88 from 8888 to 88888888, inclusive.

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Full solution

Q. Find the sum of the multiples of 88 from 8888 to 88888888, inclusive.
  1. Identify first multiple of 88: Identify the first multiple of 88 within the range.\newlineCalculation: 8888 is already a multiple of 88.
  2. Identify last multiple of 88: Identify the last multiple of 88 within the range.\newlineCalculation: 8888/8=11118888 / 8 = 1111, so the last multiple is 1111×8=88881111 \times 8 = 8888.
  3. Total multiples of 88: Determine the total number of multiples of 88 in the range.\newlineCalculation: (8888−88)/8+1=8800/8+1=1100+1=1101(8888 - 88) / 8 + 1 = 8800 / 8 + 1 = 1100 + 1 = 1101.
  4. Arithmetic series formula: Use the formula for the sum of an arithmetic series: Sn=n2×(first term+last term)S_n = \frac{n}{2} \times (\text{first term} + \text{last term}). Calculation: S1101=11012×(88+8888)S_{1101} = \frac{1101}{2} \times (88 + 8888).
  5. Calculate sum: Perform the calculation for the sum.\newlineCalculation: S1101=550.5×(8976)S_{1101} = 550.5 \times (8976).
  6. Final sum calculation: Final calculation to find the sum.\newlineCalculation: S1101=550.5×8976=4940448S_{1101} = 550.5 \times 8976 = 4940448.

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