Find the sum of the first 40 positive integers divisible by 6

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find the sum of the first 40 positive integers divisible by 6

@sorumatikbot

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Find the Sum of the First 40 Positive Integers Divisible by 6

Answer: To find the sum of the first 40 positive integers divisible by 6, we can utilize the formula for the sum of an arithmetic sequence. An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant.

Step-by-Step Solution

  1. Identify the Sequence Elements:

    • The first positive integer divisible by 6 is 6 itself.
    • The second integer in this sequence is 12 (since 6 \times 2 = 12).
    • This pattern continues, where each term is 6 \times n, where n is the position in the sequence.

  2. Arithmetic Sequence Characteristics:

    • First term (a_1): (6)
    • Common difference (d): (6)
    • We need the first 40 terms, so ( n = 40 ).

  3. Formula for the nth term of an Arithmetic Sequence:

    • The nth term (a_n) can be calculated using the formula:
      [
      a_n = a_1 + (n-1) \cdot d
      ]
    • For the 40th term:
      [
      a_{40} = 6 + (40-1) \cdot 6 = 6 + 39 \cdot 6 = 6 + 234 = 240
      ]

  4. Sum of the First n Terms of an Arithmetic Sequence:

    • The sum (S_n) of the first (n) terms is given by:
      [
      S_n = \frac{n}{2} \cdot (a_1 + a_n)
      ]

  5. Calculate the Sum:

    • Plug the known values into the formula:
      [
      S_{40} = \frac{40}{2} \cdot (6 + 240)
      ]
      [
      S_{40} = 20 \cdot 246 = 4920
      ]

Therefore, the sum of the first 40 positive integers divisible by 6 is 4920.

Summary: By identifying the sequence as an arithmetic sequence and applying the relevant formulae for the nth term and the sum of the series, we determined that the sum of the first 40 integers divisible by 6 is 4920. This solution highlights applying mathematical formulas to solve real problems conveniently and efficiently.

Would you like any further assistance with arithmetic sequences, @anonim3?