**Problem**

**Introduction**

The above was
discussed in this previous article. The earlier parts of the
problem are easy. The major sticking
point is finding the formula for

*S*. We solved that using factorisation and observation, which I feel is the best way. But what if you cannot do that and you are desperate (for example, in an exam or test)?_{n}
This article introduces Newton ’s Method, which can be used as a
back-up method, even though it is not in the regular syllabus.

**Solution**(

**Remark**

Note that number sequences in “IQ tests” (with
no problem contexts) have been debunked. In our case here, the numbers do have a
certain regularity arising from the pattern of dots. In fact this is an arithmetic
progression. What we are calculating is
the sum of an arithmetic progression. However,
Newton ’s Method
extends beyond arithmetic progressions.

**Suitable Levels**

*****

*****GCE ‘O’ Level “Elementary” Mathematics (revision)

*****GCE ‘A’ Levels H2 Mathematics (revision)

* other syllabuses that involve number
patterns and sequences

* any precocious or independent learner who
is interested

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