The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.
Arithmetic progressions: terms, sums and integer thresholds
Recover the first term and common difference from nonconsecutive terms.
Distinguish an individual term from a finite or partial sum.
Retain every positive integer solution and check threshold neighbours.
For an AP, \(u_n=a+(n-1)d\) and \(\displaystyle S_n=\frac n2[2a+(n-1)d]\). The sum from term \(p\) to term \(q\) has \(q-p+1\) terms. Indices are positive integers: test neighbouring values rather than simply rounding.