// Numbas version: exam_results_page_options {"name": "Progresiones aritm\u00e9ticas: Ecuaciones simultaneas", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "

Recall the formula for the sum of the first n terms of an arithmetic progression is \$$S_n=\\frac{n}{2}(2a+(n-1)d)\$$.

\n

The sum of the first \$$\\var{n1}\$$ terms of an arithmetic progression is \$$\\var{s1}\$$

\n

\$$\\frac{\\var{n1}}{2}(2a+\\simplify{{n1}-1}d)=\\var{s1}\$$                               equation (i)

\n

The formula for nth term of an arithmetic progression is \$$T_n=a+(n-1)d\$$.

\n

The \$$\\var{n2}th\$$ term of the same series is \$$\\var{T}\$$

\n

\$$a+\\simplify{{n2}-1}d=\\var{T}\$$                                                   equation (ii)

\n

Here we have two simultaneous equations. We can eliminate the \$$a\$$ term.

\n

\$$\\var{n1}a+\\simplify{({n1}-1)*{n1}/2}d=\\var{s1}\$$                    equation (i)

\n

\$$\\var{n1}a+\\simplify{{n1}*({n2}-1)}d=\\simplify{{n1}*{T}}\$$                  equation (ii)*\$$\\var{n1}\$$

\n

\$$\\simplify{({n1}-1)*{n1}/2-{n1}*({n2}-1)}d=\\simplify{{s1}-{n1}*{T}}\$$

\n

\$$d=\\frac{\\simplify{{s1}-{n1}*{T}}}{\\simplify{({n1}-1)*{n1}/2-{n1}*({n2}-1)}}\$$

\n

\$$d=\\simplify{({s1}-{n1}*{T})/(({n1}-1)*{n1}/2-{n1}*({n2}-1))}\$$

\n

Using this result and equation (ii) we can find the value for \$$a\$$

\n

\$$a+\\simplify{({n2}-1)*({s1}-{n1}*{T})/(({n1}-1)*{n1}/2-{n1}*({n2}-1))}=\\var{T}\$$

\n

\$$a=\\var{T}-\\simplify{({n2}-1)*({s1}-{n1}*{T})/(({n1}-1)*{n1}/2-{n1}*({n2}-1))}\$$

\n

\$$a=\\simplify{{T}-({n2}-1)*({s1}-{n1}*{T})/(({n1}-1)*{n1}/2-{n1}*({n2}-1))}\$$

\n

", "ungrouped_variables": ["n1", "s1", "n2", "T", "d", "a"], "rulesets": {}, "tags": [], "statement": "

La suma de los primeros términos \\ (\\ var {n1} \\) de una progresión aritmética es \\ (\\ var {s1} \\) y el término \\ (\\ var {n2} th \\) de la misma serie es \\ (\\ var {T} \\).

", "preamble": {"css": "", "js": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": [], "functions": {}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "

Solving arithmetic progressions using simultaneous equations