There is a staircase of 10 steps which we are to climb. We can go up one step at a time or two steps at a time, but we have to use each method at least once. (1) Suppose we can go up two steps at a time twice or more in a row. (i) If we climb the staircase going up two steps at a time just 3 times, we will go up one step at a time just $\mathbf{P}$ times, and there are $\mathbf{QR}$ different ways of climbing the staircase. (ii) If we can go up two steps at a time twice or more in a row, there are altogether $\mathbf{ST}$ different ways of climbing the staircase. (2) Suppose we cannot go up two steps at a time twice or more in a row. (i) If we climb the staircase going up two steps at a time just twice, we will go up one step at a time just $\mathbf{U}$ times, and there are $\mathbf{VW}$ different ways of climbing the staircase. (ii) If we cannot go up two steps at a time twice or more in a row, there are altogether $\mathbf{XY}$ different ways of climbing the staircase.
There is a staircase of 10 steps which we are to climb. We can go up one step at a time or two steps at a time, but we have to use each method at least once.
(1) Suppose we can go up two steps at a time twice or more in a row.
(i) If we climb the staircase going up two steps at a time just 3 times, we will go up one step at a time just $\mathbf{P}$ times, and there are $\mathbf{QR}$ different ways of climbing the staircase.
(ii) If we can go up two steps at a time twice or more in a row, there are altogether $\mathbf{ST}$ different ways of climbing the staircase.
(2) Suppose we cannot go up two steps at a time twice or more in a row.
(i) If we climb the staircase going up two steps at a time just twice, we will go up one step at a time just $\mathbf{U}$ times, and there are $\mathbf{VW}$ different ways of climbing the staircase.
(ii) If we cannot go up two steps at a time twice or more in a row, there are altogether $\mathbf{XY}$ different ways of climbing the staircase.