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  "body": "\n## Example\n\nLet $a$ and $d$ be real numbers.\nThe sequence $(a_n)_{n \\ge 0}$ with $a_n = a + n d$ is the *arithmetic progression* with first term $a$ and common difference $d$.\nIt is a [[sequence|0G9EKM0E15]] in which each term minus the previous one is the same number: $a_{n+1} - a_n = d$ for every $n \\ge 0$.\nFor example, $a = 3$ and $d = 4$ give $3, 7, 11, 15, \\ldots$.\n\nThe sum of the first $n$ terms is\n$$a_0 + a_1 + \\cdots + a_{n-1} = n a + \\frac{n(n-1)}{2} d = \\frac{n (a_0 + a_{n-1})}{2}.$$\nCheck: $\\sum_{k=0}^{n-1} (a + kd) = na + d \\sum_{k=0}^{n-1} k$, and $\\sum_{k=0}^{n-1} k = n(n-1)/2$ by the [[sum of the first natural numbers|01QZNHQWYD]].\nFor $3 + 7 + 11 + 15$ this gives $4 \\cdot (3 + 15) / 2 = 36$.\n\n## Remarks\n\nThe progression converges if and only if $d = 0$, since otherwise $\\abs{a_n} \\ge n \\abs{d} - \\abs{a}$ tends to $\\infty$.\nSo the series $\\sum_n a_n$ of an arithmetic progression diverges unless $a = d = 0$.\nEach term is the average of its two neighbours: $a_n = (a_{n-1} + a_{n+1})/2$ for $n \\ge 1$.\n\n## Explanations\n\n### L1 human\n\nAn arithmetic progression goes up or down in equal steps, like $2, 5, 8, 11$.\nTo add its terms, pair the first with the last, the second with the second to last, and so on.\nEvery pair has the same sum, so the total is the number of terms times the average of the first and the last term.\n",
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