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Find the sum of all 7 multiples lies between 17 to 549
The sum of all multiples of 7 between 17 and 549 is 21,546.
To find the sum of all multiples of 7 between 17 and 549, we need to find the first multiple of 7 within this range and the last multiple of 7 within this range, and then calculate the sum of the arithmetic sequence formed by these multiples.
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First, let's find the first multiple of 7 within the range:
17 ÷ 7 = 2 remainder 3
So, the first multiple of 7 within the range is 7 × 3 = 21.
Next, let's find the last multiple of 7 within the range:
549 ÷ 7 = 78 remainder 3
So, the last multiple of 7 within the range is 7 × 78 = 546.
Now, we have the first multiple (21) and the last multiple (546) within the range. We can calculate the sum of the arithmetic sequence using the formula:
S = (n/2)(a1 + an)
Where:
- S is the sum of the arithmetic sequence
- n is the number of terms in the sequence
- a1 is the first term
- an is the last term
To find n, we can use the formula for the nth term of an arithmetic sequence:
an = a1 + (n - 1)d
Where d is the common difference (which is 7 in this case).
So, we can find n as follows:
546 = 21 + (n - 1) × 7 546 - 21 = 7n - 7 525 = 7n - 7 7n = 532 n = 532 ÷ 7 n = 76
Now that we have n, we can calculate the sum S:
S = (76/2)(21 + 546) S = 38 × 567 S = 21546
So, the sum of all multiples of 7 between 17 and 549 is 21,546.
Arithmetic Sequences and Series
Arithmetic Sequence: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the "common difference." For example, 2, 5, 8, 11, 14 is an arithmetic sequence with a common difference of 3.
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The general form of an arithmetic sequence can be represented as: a_n = a_1 + (n - 1) * d Where:
- a_n is the nth term of the sequence.
- a_1 is the first term of the sequence.
- d is the common difference.
- n is the term number.
Arithmetic Series: An arithmetic series is the sum of the terms of an arithmetic sequence. For example, the sum of the first n terms of the arithmetic sequence 2, 5, 8, 11, 14 would be an arithmetic series.
The sum of the first n terms of an arithmetic series can be calculated using the formula: S_n = (n / 2) * (a_1 + a_n) Where:
- S_n is the sum of the first n terms.
- n is the number of terms.
- a_1 is the first term.
- a_n is the nth term.
Another common formula for the sum of an arithmetic series is:
S_n = (n / 2) * (2 * a_1 + (n - 1) * d)
Examples:
Find the 10th term of the arithmetic sequence 3, 7, 11, 15, ...whered = 4.
a_10 = 3 + (10 - 1) * 4 = 3 + 9 * 4 = 3 + 36 = 39
Find the sum of the first 20 terms of the arithmetic sequence 2, 5, 8, 11, ... where d = 3.
S_20 = (20 / 2) * (2 + (20 - 1) * 3) = 10 * (2 + 57) = 10 * 59 = 590
Arithmetic sequences and series have various applications in real-world problems, such as calculating finances, analyzing patterns, and modeling physical phenomena.
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