[Math] Common difference between terms of the arithmetic progression

algebra-precalculussequences-and-series

In an finite arithmetic progression: $S_5=55$, sum of the last five terms is 215 and total sum is $S_n=351$ . What is common difference between terms of the arithmetic progression? Why?

Best Answer

HINT: Let $a_1$ be the first term and $a_n$ the last term, and let $d$ be the common difference.

  • Show that the sum of the first five terms is $5a_1+10d$, and conclude that $a_1+2d=11$.
  • Show that the sum of the last five terms is $5a_n-10d$, and conclude that $a_n-2d=43$.
  • What is $a_1+a_n$?
  • Use the fact that $S_n=\frac{n(a_1+a_n)}2$ to find $n$.
  • Then use the fact that $a_n=a_1+(n-1)d$ to get an equation in the unknowns $a_1$ and $d$ that you can pair with $a_1+2d=11$ to solve for $d$.
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