Aussie Maths & Science Teachers: Save your time with SmarterEd
A toy train track consists of a number of pieces of track which join together.
The shortest piece of the track is 15 centimetres long and each piece of track after the shortest is 2 centimetres longer than the previous piece.
The total length of the complete track is 7.35 metres.
The length of the longest piece of track, in centimetres, is
A. `21`
B. `47`
C. `49`
D. `55`
E. `57`
`D`
`text(Sequence is 15, 17, 19 , . . .)`
| `text(AP where)\ \ \ a` | `= 15, and` |
| `d` | `= 2` |
| `S_n` | `= n / 2 [ 2a + (n – 1) d ] = 735\ text(cm)` |
| `:. 735` | `= n / 2 [ 2 xx 15 + (n – 1) 2 ]` |
| `= n / 2 [ 30 + 2n – 2 ]` | |
| `= n^2 + 14n` | |
| `0` | `= n^2 + 14n – 735` |
`text(Using the quadratic formula)`
| `n` | `= {–14 +- sqrt (14^2 – 4. 1. (–735))} / (2 xx 1)` |
| `= (–14 +-56) / 2` | |
| `= 21 \ \ \ \ (n > 0)` |
| `:.\ text(Longest piece of track)` | `= a + (n – 1) d` |
| `= 15 + (21 – 1) 2` | |
| `= 55\ text(cm)` |
`=> D`