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.

Interpreting sequence models

TGM Original

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 \(d=\dfrac{160-102}{3}=\dfrac{58}{3}\), 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: \(6-3+\dfrac32-\cdots\) has sum \(\dfrac6{1+\dfrac12}=4\). 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.

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