| 2n +1 | 1+3+5+....+(2n−1) | |||
Dany jest ciąg o wyrazie ogólnym an = | − | |||
| 2 | n+1 |
ODP:
ciąg rosnący
| 2 n + 1 | n2 | (2n+1)*( n + 1) − n2 | ||||
an = | − | = | = | |||
| 2 | n + 1 | 2n + 2 |
| 2n2 +2n + n + 1 − n2 | n2 + 3 n + 1 | |||
= | = | |||
| 2 n + 2 | 2n + 2 |
| ( n +1)2 + 3*( n +1) + 1 | ||
an +1 = | = | |
| 2*( n +1) + 2 |
| n2 + 2n + 1 + 3n + 3 + + 1 | n2 +5n + 5 | |||
= | = | |||
| 2n + 4 | 2n + 4 |
| n2 + 5n + 5 | n2 +3n + 1 | |||
an+1 − an = | − | = | ||
| 2n + 4 | 2n + 2 |
| 2*(n2+3n +3) | ||
= ... = | > 0 | |
| (2n+4)*(2n + 2) |
| (2n + 1)*(n + 1) − 2n2 | ||
an = ... = | = ... | |
| 2n + 2 |