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  "body": "\n## Definition\n\nLet $a$ and $r$ be real numbers.\nThe geometric sequence with initial term $a$ and common ratio $r$ is the [[real sequence|0CTV783HYT]] $(a\\,r^n)_{n \\in \\mathbb{N}}$.\nIts terms are $a, a r, a r^2, a r^3, \\dots$, and each term is $r$ times the term before it.\n\n## Remarks\n\nMany texts require $a \\ne 0$ and $r \\ne 0$, so that the quotient of two consecutive terms is always $r$.\nThe sequence $1, 2, 4, 8, \\dots$ has $a = 1$ and $r = 2$. The sequence $1, 1/2, 1/4, \\dots$ has $r = 1/2$.\nFor $|r| \u003c 1$, the sequence [[converges to $0$|0Q7K4AXE53]]. For $r \u003e 1$ and $a \u003e 0$, it [[tends to infinity|0TZK91NQE3]].\nThe sum of the terms is the [[geometric series|01X5CVV1Q5]].\n\n## Explanations\n\n### L1 human\n\nIn a geometric sequence, you go from one term to the next by multiplication with the same number $r$.\nIf $r = 3$ and you start at $2$, you get $2, 6, 18, 54, \\dots$.\n",
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