examday

/TMUA

An arithmetic sequence $T$ has first term $a$ and common difference $d$, where $a$ and $d$ are non-zero integers.

Property $P$ is:

For some positive integer $m$, the sum of the first $m$ terms of the sequence is equal to the sum of the first $2m$ terms of the sequence.

For example, when $a=11$ and $d=-2$, the sequence $T$ has property $P$, because

i.e. the sum of the first 4 terms equals the sum of the first 8 terms.

Which of the following statements is/are true?

• I For $T$ to have property $P$, it is sufficient that $ad<0$.
• II For $T$ to have property $P$, it is necessary that $d$ is even.