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When I learned mathematics at school in England in around 1970 we were taught mathematical series, rather than sequences. The two words must mean the same thing. I would very much like to know if there is an international convention in maths like there is in chem or phys.
A sequence where consecutive terms have a common difference is called an arithmetic sequence.
Here we know the common difference is 2.
The nth term of any arithmetic sequence is given by a + (n - 1)*d where a is the first term of the sequence and d is the common difference.
The 6th and 7th terms of the sequence that we need are given as a + 5d and a + 6d.
Since the difference between 2 consecutive terms is always 2, then the sequence is an arithmetic sequence.
To determine any term of an arithmetic sequence, we'll have to know 2 basic terms: the 1st term and the common difference. The common difference is 2 but the 1st term is not known.
Since the problem does not provide other constraint, we'll impose the following:
a2 + a3 = 8 (1)
By definition, the difference between 2 consecutive terms of an arithmetial progression is the common difference of the arithmetic sequence.
a3 - a2 = d
But, the common difference is d = 2, then:
a3 - a2 = 2 (2)
We'll add (1) + (2):
a2 + a3 + a3 - a2 = 8 + 2
We'll eliminate and combine like terms:
2a3 = 10
a3 = 10/2
a3 = 5
a2 = a3 - 2
a2 = 5 - 2
a2 = 3
But a2 = a1 + d
3 = a1 + 2
a1 = 3 - 2
a1 = 1
Since we know a1 and d, we'll determine a6 and a7:
a6 = a1 + 5d
a6 = 1 + 10
a6 = 11
a7 = 13
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