Suatu barisan aritmatika memiliki \(U_5 = 13\) dan \(U_{13} = 29\). Jumlah dua puluh lima suku pertama barisan tersebut adalah....

diketahui:

\(\begin{aligned} U_5 &= 13 \\ U_{13} &= 29 \end{aligned}\)

ditanya:

\(S_{25}?\)

Penyelesaian

mencari nilai b

\(\begin{aligned} b &= \frac{U_n - U_x}{n-x} \\ &= \frac{U_{13} - U_5}{13-5} \\ &= \frac{29-13}{8} \\ &= \frac{16}{8} = 2 \end{aligned}\)

mencari nilai a

\(\begin{aligned} U_n &= a+(n-1)b \\ U_5 &= a+(5-1)2 \\ 13 &= a+(4)2 \\ a &= 13-8 = 5 \end{aligned}\)

mencari nilai \(U_{25}\)

\(\begin{aligned} S_{n} &= \frac{n}{2}(2a + (n-1)b) \\ S_{25} &= \frac{25}{2}(2(5) + (25-1)2) \\ &= \frac{25}{2}(10+(24)2) \\ &= \frac{25}{2}(10+48) \\ &= \frac{25}{\cancel{2}}\times \cancel{58}^{\ 29} \\ &= 25\times 29 \\ &= 725 \end{aligned}\)