Enter the count and base number to quickly calculate the sum of the first N multiples of the base number.
The sum of the first N multiples can be seen as the sum of an arithmetic sequence. The multiples themselves form an arithmetic sequence, where the common difference equals the base number. Therefore, we can use the Arithmetic Sequence Sum Formula to calculate the sum.
For an arithmetic sequence with the first term \( a_1 \) and the \( n \)-th term \( a_n \), the sum \( S_n \) is given by:
\( S_n = \frac{n}{2} \times (a_1 + a_n) \)
where:
- \( S_n \) is the sum of the multiples,
- \( n \) is the number of multiples,
- \( a_1 \) is the first multiple (which is the base number itself),
- \( a_n \) is the \( n \)-th multiple (which is \( b \times n \)).
Solution:
1. Find the first and 15th multiples:
\( a_1 = 7 \)
\( a_{15} = 7 \times 15 = 105 \)
2. Calculate the sum using the arithmetic sequence formula:
\( S_{15} = \frac{15}{2} \times (7 + 105) = \frac{15}{2} \times 112 = 15 \times 56 = 840 \)
Result: 840
3. Verification:
Find the first 15 multiples and manually calculate the sum:
The first 15 multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105.
Manually summing them:
\( 7 + 14 + 21 + 28 + 35 + 42 + 49 + 56 + 63 + 70 + 77 + 84 + 91 + 98 + 105 = 840 \)
The formula result matches the manual calculation.
Solution:
1. Find the first and 10th multiples:
\( a_1 = 3 \)
\( a_{10} = 3 \times 10 = 30 \)
2. Calculate the sum using the arithmetic sequence formula:
\( S_{10} = \frac{10}{2} \times (3 + 30) = 5 \times 33 = 165 \)
Result: 165
3. Verification:
Find the first 10 multiples and manually calculate the sum:
The first 10 multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30. Their sum is:
\( 3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 + 30 = 165 \)
Verified Result: 165
The formula result matches the manual calculation.