Term counts are positive integers. For a term use arn-1; for a total use a(1-rn)/(1-r). Logarithms undo powers, but ln r<0 when 0<r<1, so dividing an inequality by ln r reverses its sign. Isolate the power; take logarithms if needed; solve with the correct inequality; choose the integer that satisfies the actual condition; check it and the preceding index. For u_n=100(0.8)n-1, u_n<50 gives n-1>ln(0.5)/ln(0.8) 3.106. Term 5 is first: u_4=51.2, u_5=40.96. Exam check: rounding to the nearest integer is not a threshold method. For “at least”, equality is allowed; for “exceeds”, it is not.

Term counts are positive integers. For a term use \(ar^{n-1}\); for a total use \(\dfrac{a(1-r^n)}{1-r}\).

Logarithms undo powers, but \(\ln r<0\) when \(0<r<1\), so dividing an inequality by \(\ln r\) reverses its sign.

Isolate the power; take logarithms if needed; solve with the correct inequality; choose the integer that satisfies the actual condition; check it and the preceding index.

For \(u_n=100(0.8)^{n-1}\), \(u_n<50\) gives \(n-1>\dfrac{\ln(0.5)}{\ln(0.8)}\approx3.106\). Term 5 is first: \(u_4=51.2\), \(u_5=40.96\).

Exam check: rounding to the nearest integer is not a threshold method. For “at least”, equality is allowed; for “exceeds”, it is not.

Geometric indices and thresholds

Geometric indices and thresholds