Interpreting sequence models

Choose the index before substituting. If u_1 is the initial quantity, the amount after t changes is u_t+1; if the model starts at u_0, it is u_t. **Model** | **Diagnostic** | **Interpretation** Arithmetic | u_n+1-u_n=d | A constant amount is added at each step. Geometric | u_n+1=ru_n | The same multiplier is applied at each step. Approximate model | Compare several successive differences or ratios. | Real measurements may fluctuate; a fitted pattern supports estimates, not a guarantee. Example Visitor counts 102,119,143,160 have differences 17,24,17. An approximate arithmetic model uses d= 160-1023= 583, predicting about 179 at the next observation. State that the increase is assumed to remain similar. For a constant geometric sequence with r=1, use u_n=a and S_n=na; the usual quotient formula for S_n would divide by zero. For a nonzero first term, an infinite geometric series converges exactly when |r|<1. Negative ratios can converge: 6-3+ 32-. s has sum 61+ 12=4. At r=-1, partial sums oscillate rather than approach a limit. For a threshold, solve for the index and test neighbouring integer indices in the original model. Check whether the question asks for one term or the total of several terms.