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The first three terms of a geometric sequence are $u_1 =$ {u_1}, $u_2 =$ {u_2} and $u_3 =$ {u_3}.

", "advice": "

In part (a) the common ration can be found by dividing a term by the preceding term. Either $\\frac{u_3}{u_2}$ or $\\frac{u_2}{u_1}$

$r = 1 \\div$ {reciprocal_r}

In part (b) use the formula $u_n = u_1r^{n-1}$ and substitute value for $r$ and $u_1$ and set $u_n = 2$

2 = {u_1}/{reciprocal_r}$^{n-1}$

{reciprocal_r}$^{n-1} =$ {u_1} $\\div 2 =$ {half_u_1}

This is a 'nice' power of {reciprocal_r}

$n - 1 = 5$

$n = 6$

In part (c) use the formula $S_n = \\frac{u_1(1-r^n)}{1-r}$ it is preferable in this case since $|r|<1$

substituting values that you know gives sum of first {how_many} terms to be {sum_c}

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Third term of the geometric sequence

", "templateType": "anything"}, "reciprocal_r": {"name": "reciprocal_r", "group": "Ungrouped variables", "definition": "random(2 .. 6#1)", "description": "

reciprocal of the common ratio.

", "templateType": "randrange"}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "1/{reciprocal_r}", "description": "

common ratio, set to be a fraction either $\\frac{1}{2}, \\frac{1}{3}, \\frac{1}{4}, \\frac{1}{5}$ or $\\frac{1}{6}$

", "templateType": "anything"}, "u_2": {"name": "u_2", "group": "Ungrouped variables", "definition": "{u_3}*{reciprocal_r}", "description": "

Second term of the geometric sequence

", "templateType": "anything"}, "u_1": {"name": "u_1", "group": "Ungrouped variables", "definition": "{u_2}*{reciprocal_r}", "description": "

First term of the geometric sequence

", "templateType": "anything"}, "u_6": {"name": "u_6", "group": "Ungrouped variables", "definition": "2", "description": "

Sixth term of geometric sequence

", "templateType": "number"}, "how_many": {"name": "how_many", "group": "Ungrouped variables", "definition": "random(20 .. 30#5)", "description": "

Variable to determine how many terms should be summed in part (c)

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Answer to part (c), summing appropriate number of terms

", "templateType": "anything"}, "half_u_1": {"name": "half_u_1", "group": "Ungrouped variables", "definition": "u_1/2", "description": "

u_1 divided by u_6 required in advice.

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Find the value of $r$ the common ratio.

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Find the value of $n$ for which $u_n =$ {u_6}.

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Find the sum of the first {how_many} terms of the sequence.

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