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Part a)
\nTo find the first term $a_1$, we need to first start with the formula for the $n^{th}$ term:
\n$a_n=a_1+d(n-1)$ $_{..(i)}$ which is rearranged in terms of $a_1$: $a_1=a_n-d(n-1)$ $_{..(ii)}$
\nWe then identify the variables which we know, namely: $n=\\var{n}$, $a_n=\\simplify{{t1}+({n}-1)*{d}}$, $d=\\var{d}$
\nThese values can then be subsituted into $_{(ii)}$ and solved for $a_1$:
\n$a_1=\\simplify{{t1}+({n}-1)*{d}}-\\var{d}(\\var{n}-1)$
\nPart b)
\nTo find the sum of the first $\\var{tsum}$ terms we take the formula for the sum of an arithmetic series:
\n$S_n=\\frac{n}{2}(a_1+a_n)$ $_{..(iii)}$
\nWhile we don't yet know the value of $a_n$, it is obtainable with formula $_{..(i)}$. It is common to substitute this formula into the formula for the sum as follows:
\n$S_n=\\frac{n}{2}(a_1+a_1+d(n-1))$
\n$\\therefore\\;\\;\\;S_n=\\frac{n}{2}(2a_1+d(n-1))$ $_{..(iv)}$
\nWe identify the variables which we know, namely: $n=\\var{tsum}$, $a_1=\\var{t1}$, $d=\\var{d}$
\nThey are substituted into $_{..(iv)}$ and solved for $S_n$:
\n$S_\\var{tsum}=\\frac{\\var{tsum}}{2}(2\\times\\var{t1}+\\var{d}(\\var{tsum}-1))$
\nPart c)
\nTo determine the least value of $n$ for which the sum of the first $n$ terms of the series exceeds $1000$, the sum formula $_{..(iv)}$ is set up with the lower limit value of $1000$ substituted in:
\n$1000=\\frac{n}{2}(2\\times\\var{t1}+\\var{d}(n-1))$
\nIt is then expanded algebraically to give the following quadratic:
\n$\\simplify{{d}/2}n^2+\\simplify{{2*{t1}-{d}}/2}n-1000=0$
\n$n$ can then be found using the quadratic formula:
\n$n=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$ where $a=\\simplify{{d}/2}$, $b=\\simplify{{2*{t1}-{d}}/2}$ and $c=-1000$
\nThus,
\n$n=\\frac{-\\var{b}\\pm\\sqrt{(\\var{b})^2-\\simplify{{-1000*4*{a}}}}}{\\simplify{2*{a}}}$
\n$\\therefore\\;\\;\\;n=-\\left(\\simplify{{2*{t1}-{d}}/{2*2*{a}}}\\right)\\pm\\left(\\frac{\\sqrt{\\simplify{{bsq}-{-1000*4*{a}}}}}{\\simplify{2*{a}}}\\right)$
\nIn this context, we are talking of positive $n^{th}$ terms, so we disregard the negative square root.
\nThe result is then computed to be: $\\var{nc}$
\nIf this is the value of $n$ which produces $1000$, it follows that the next whole number will be the least value of $n$ for which the sum of the first $n$ terms of the series exceeds $1000$.
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\nThe common difference is $\\var{d}$
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