The first four terms of a sequence are $2,\ 3,\ 4,\ 5.$ Find an expression, in terms of $n$, for

Question

The first four terms of a sequence are $2,\ 3,\ 4,\ 5.$ Find an expression, in terms of $n$, for its $n$th term. [1] Cans are stacked in triangular arrangements. Each new row requires more cans, as shown in the table. Stage Calculation $1$$1+2=3$ $2$$3+3=6$ $3$$6+4=10$ $4$$10+5=15$ $10$ $n$ The numbers $1,\ 3,\ 6,\ 10,\ldots$ are triangular numbers. Complete the entries for Stage $10$ and Stage $n$. [3] Explain whether $300$ occurs as one of the new totals in this pattern. [1] The difference between two consecutive totals in the pattern is $31$. Find these two totals. [2]

Accepted Answer

(a) $n+1$ (b)(i) $55+11=66$ and $\frac{n(n+1)}2+(n+1) =\frac{(n+1)(n+2)}2$ (ii) Yes (iii) $465$ and $496$ ​

Stage	Calculation
\(1\)	\(1+2=3\)
\(2\)	\(3+3=6\)
\(3\)	\(6+4=10\)
\(4\)	\(10+5=15\)
\(10\)
\(n\)

Number Patterns - General Term of a Sequence

Number Patterns - General Term of a Sequence

Solution:

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