The difference between the sum of the first 20 terms of the arithmetic sequence 6, 10, 14 and the arithmetic sequence 15, 19, 23 is 180.
Here's how we determine that:
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Understanding Arithmetic Sequences
An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference.- For the sequence 6, 10, 14, the common difference is 4 (10-6 = 4, 14-10 = 4).
- For the sequence 15, 19, 23, the common difference is 4 (19-15 = 4, 23-19 = 4).
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Calculating the Sum of an Arithmetic Sequence
The sum (S) of the first n terms of an arithmetic sequence is given by the formula:
S = (n/2) * [2a + (n-1)d]
where:- n is the number of terms
- a is the first term
- d is the common difference
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Applying the Formula to Each Sequence
- Sequence 1: 6, 10, 14
- n = 20
- a = 6
- d = 4
- S1 = (20/2) [2(6) + (20-1)4] = 10 [12 + 19 4] = 10 [12 + 76] = 10 * 88 = 880
- Sequence 2: 15, 19, 23
- n = 20
- a = 15
- d = 4
- S2 = (20/2) [2(15) + (20-1)4] = 10 [30 + 19 4] = 10 [30 + 76] = 10 * 106 = 1060
- Sequence 1: 6, 10, 14
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Finding the Difference
The difference between the sums of the first 20 terms of the two sequences is |S2 - S1| = |1060 - 880| = 180
Therefore, the difference between the sums of the first 20 terms is 180. This matches the result from the provided reference.
