Question

The Arroyos are planning to build a brick patio that approximates the shape of a trapezoid. The shorter base of the trapezoid needs to start with a row of 5 bricks, and each row must increased by 2 bricks on each side until there are 25 rows. How many bricks do the Arroyos need to buy? a. 1325 bricks c. 3125 bricks b. 1850 bricks d. 1575 bricks

Answer 1

so they start off with 5 bricks, then they add 2 bricks on the left side and 2 bricks on the right side, namely 4 bricks, so the first row is 5 bricks, the next row is 5+4 or 9 bricks and so on.

5, 9, 13, 17.... <--- as you can see the "common difference" is 4.


`\bf n^(th)\textit{ term of an arithmetic sequence} \\\\ a_n=a_1+(n-1)d\qquad \begin{cases} n=n^(th)\ term\\ a_1=\textit{first term's value}\\ d=\textit{common difference}\\ ----------\\ a_1=5\\ d=4\\ n=25 \end{cases} \\\\\\ a_(25)=5+(25-1)(4)\implies a_(25)=5+(24)(4) \\\\\\ a_(25)=5+96\implies a_(25)=101\\\\ -------------------------------`


`\bf \textit{ sum of a finite arithmetic sequence} \\\\ S_n=\cfrac{n(a_1+a_n)}{2}\qquad \begin{cases} n=n^(th)\ term\\ a_1=\textit{first term's value}\\ ----------\\ a_1=5\\ a_(25)=101\\ n=25 \end{cases} \\\\\\ S_(25)=\cfrac{25(5+101)}{2}\implies S_(25)=\cfrac{25(106)}{2}\implies S_(25)=1325`

Answer 2

The answer would be a. 1325 bricks.
COmmon difference= 4Use arithmetic sequence:An=A1_(n-1)dA25=5_(25-1)(4)A25=5(24)(4)A25=5+96A25=101
Sn=n(A1+An)/2S25=25(5+101)/2=25(106)/2=1325