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A hall has 10 rows of seats. Row 1 has 16 seats. Each row adds 3 more. How many seats does the hall hold in all?

What this track covers

This track starts from the term rule, aₙ = a + (n − 1)d. Here a is the first term and d is the step. The letter n is the place of a term. Also, aₙ is the term at place n. Each term is the one before plus the same step. Such a list is an arithmetic progression, or AP. This track uses the rule, then adds up all the terms.

• Every end pair — pair the first row with the last.
• Total from a and d — no last term needed.
• Work it backwards — the total is known; find n or d.
• Terms from the sums — find the list from its totals.
• Two sums name it — two totals fix a whole list.
• Stacks and rows — real piles and halls, added up.
• When it slides — a step can be negative too.
• Which question wants the sum — what is given, what is missing.
• Is it a term of the list? — n must be whole.
• Two facts fix the list — two terms give a and d.

Quick catch-up

• The same step every time — the term rule, aₙ = a + (n − 1)d.
• Adding the first n — pairing the counting numbers, Class 9.
• A rule that jumps straight there — any term in one line.
• A rule that looks back — each term built from the one before.
• Scale the equations, then take one away — elimination, Class 9.
• The pair with two jobs — splitting a middle term.

Chapter 5 of your Mathematics book is Arithmetic Progressions. You can find any term. Now you learn to add them all up.