A geometric progression is generated by multiplication by one fixed number: u_n+1=ru_n. A ratio test by division is valid only when the divisor is nonzero. For a 0,r=0, the terms are a,0,0,. Repeated multiplication is well defined even though 0/0 is not. Check the generating rule when zeros occur; otherwise compare successive nonzero ratios. For an AP, compare differences even when a term is zero. 5,0,0, generated by multiplication by zero is a GP with r=0; its total is 5. It is not an AP because the first differences are -5,0. Exam check: the all-zero sequence has d=0, and every multiplicative ratio generates it. Do not claim a uniquely determined ratio from 0/0.

A geometric progression is generated by multiplication by one fixed number: \(u_{n+1}=ru_n\). A ratio test by division is valid only when the divisor is nonzero.

For \(a\ne0,r=0\), the terms are \(a,0,0,\ldots\). Repeated multiplication is well defined even though \(\dfrac{0}{0}\) is not.

Check the generating rule when zeros occur; otherwise compare successive nonzero ratios. For an AP, compare differences even when a term is zero.

\(5,0,0,\ldots\) generated by multiplication by zero is a GP with \(r=0\); its total is \(5\). It is not an AP because the first differences are \(-5,0\).

Exam check: the all-zero sequence has \(d=0\), and every multiplicative ratio generates it. Do not claim a uniquely determined ratio from \(\dfrac{0}{0}\).

Zero terms and multiplicative rules

Zero terms and multiplicative rules