Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.2 Arithmetic Sequences - 12.2 Assess Your Understanding - Page 815: 68

$30$ rows

Consider the arithmetic sequence: $S_n=2040$ $a_1=10$ $d=4$ Determine the number of rows $n$ solving the equation: $S_n=\dfrac{n(2a_1+(n-1)d)}{2}$ $2040=\dfrac{n(2(10)+(n-1)(4))}{2}$ $4080=n(20+4n-4)$ $4080=4n^2+16n$ $4n^2+16n-4080=0$ $4(n^2+4n-1020)=0$ $n^2+4n-1020=0$ $n^2+4n+4-1020-4=0$ $(n+2)^2-1024=0$ $(n+2-32)(n+2+32)=0$ $(n-30)(n+34)=0$ $n-30=0\Rightarrow n=30$ $n+34=0\Rightarrow n=-34$ As the number of rows must be positive, the only solution is: $n=30$