TMUA Sequences and Series

Sequences and series on TMUA Paper 1 split into three families, and once you can name which family the question belongs to, the formulas do the rest. The difficulty in these problems usually sits in the setup, not the algebra.

• Arithmetic. Term: $a_n = a + (n-1)d$. Sum: $S_n = \tfrac{n}{2}(2a + (n-1)d)$. Most TMUA AP questions give you two facts about specific terms and ask you to recover $a$ and $d$.
• Geometric. Term: $a_n = ar^{n-1}$. Sum: $S_n = a(1-r^n)/(1-r)$. Sum to infinity exists iff $|r| < 1$ and equals $a/(1-r)$. The trap is forgetting the $|r| < 1$ condition when the question asks “does the sum to infinity exist?”.
• Binomial expansion (positive integer $n$). $(a + b)^n = \sum \binom{n}{k} a^{n-k} b^k$. TMUA picks out single coefficients or asks which terms are divisible by something — the binomial coefficient identities matter more than computing the whole expansion.
• Iterative sequences. $a_{n+1} = f(a_n)$ with a starting value. Fixed points satisfy $a = f(a)$. The behaviour around a fixed point depends on $|f'(a)|$.

The move. Identify the family in the first read. Arithmetic has a constant gap; geometric has a constant ratio. The question won’t tell you which; it will give you facts and expect you to choose the model.