In a series of positive numbers, t_𝑛+1 + t_𝑛 = S, t_𝑛+1﹣t_𝑛 = D and t_𝑛+2 = t_𝑛+1 + S﹣D, where 𝑛 is a natural number. If t₁ = 2 and t₂ = 4, then find the sum of the first ten terms of the series.

Using the first two terms, the common sum S is 2 plus 4, which equals 6, and the common difference D is 4 minus 2, which equals 2. The next term is found by adding the previous term to S minus D, meaning we add 6 minus 2, or 4, to each preceding term. This makes the series 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, which is a geometric progression with a first term of 2 and a common ratio of 2. The sum of the first ten terms of this progression is 2 multiplied by (2^10 minus 1), which gives a total of 1023.

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