Porisma

Worked solution

Check a GP proof and a constant sequence

(a) Why is showing only u2/u1=u3/u2u_2/u_1=u_3/u_2 insufficient to conclude that a sequence is a GP? (b) Is the constant sequence 2,2,2,…,22,2,2,\ldots,2 an AP or a GP?

CJC · Sequences and Series · Unnumbered GP and constant-sequence questions, page 17

A test for every neighbouring pair

Answer

Two ratios alone do not prove that every consecutive ratio is the same.

Use the general neighbouring terms

un+1un=rfor every valid positive integer n\frac{u_{n+1}}{u_n}=r\qquad\text{for every valid positive integer }n

The printed challenge refers back to the AP proof warning. Under the lecture’s nonzero-term convention, the general ratio must be constant for all valid indices.

How did this part go?

The constant sequence of twos

Answer

Both: AP with a=2,d=0a=2,d=0; GP with a=2,r=1a=2,r=1.

Check differences and ratios

2−2=0,22=12-2=0,\qquad\frac22=1

Every valid neighbouring pair gives the same difference and ratio.

How did this part go?

Solution notes

A zero constant sequence is an AP with d=0 but is excluded from GP by this lecture’s nonzero-term definition.