nth term of linear sequences Worked Solutions — GCSE Maths

Find the gaps between consecutive terms

Subtract each term from the one after it.

Check the gaps are all the same

Every gap is $2$, so the sequence is linear and the common difference is $2$.

State the common difference

The terms increase by the same amount each time.

Find the gaps between consecutive terms

Subtract each term from the one after it.

Check the gaps are all the same

Every gap is $-3$, so the sequence is linear and the common difference is $-3$.

State the common difference

The terms are decreasing, so the common difference is negative.

Find the common difference

The gap between one term and the next is the same every time, so this is a linear (arithmetic) sequence. It goes up in $2s$, so the common difference is $d = 2$.

Write down the multiples of the common difference

Because the common difference is $2$, the $n\mathrm{th}$ term must start with $2n$. Working out $2n$ for $n = 1$, $2$, $3$, $4$ gives $2$, $4$, $6$, $8$.

Write down the $n\mathrm{th}$ term

Putting the $2n$ and the constant together gives the rule $2n$.

Find the common difference

The gap between one term and the next is the same every time, so this is a linear (arithmetic) sequence. It goes up in $5s$, so the common difference is $d = 5$.

Write down the multiples of the common difference

Because the common difference is $5$, the $n\mathrm{th}$ term must start with $5n$. Working out $5n$ for $n = 1$, $2$, $3$, $4$ gives $5$, $10$, $15$, $20$.

Write down the $n\mathrm{th}$ term

Putting the $5n$ and the constant together gives the rule $5n$.

Find the common difference

The gap between one term and the next is the same every time, so this is a linear (arithmetic) sequence. It goes up in $1s$, so the common difference is $d = 1$.

Write down the multiples of the common difference

Because the common difference is 1, the $n\mathrm{th}$ term must start with n. Working out n for $n = 1$, $2$, $3$, $4$ gives $1$, $2$, $3$, $4$.

Adjust by the constant and write the $n\mathrm{th}$ term

Each term of the sequence is 3 more than the matching multiple of 1, so the $n\mathrm{th}$ term is n + 3. Check $n = 1$: $1 + 3 = 1 + 3 = 4$, the first term.