The nth term questions with answers

Questions

1. Question 1nth term, going up

   Find an expression, in terms of $n$, for the $n$th term of the sequence $1, \ 4, \ 7, \ 10, \ \ldots$.

   Show the worked solution

   1. Step 1: Find the common difference: it is the number in front of n$\mathstrut 4 - 1 =$ $3$
   2. Step 2: Compare with (difference)n, which starts at the difference: first term − difference = the number to add$\mathstrut 1 - 3 =$ $-2$
   3. Step 3: So the nth term is (difference)n + (the number to add)$T_n = 3n - 2$
2. Question 2nth term, going down

   Find an expression, in terms of $n$, for the $n$th term of the sequence $40, \ 35, \ 30, \ 25, \ \ldots$.

   Show the worked solution

   1. Step 1: Find the common difference: it is the number in front of n$\mathstrut 35 - 40 =$ $-5$
   2. Step 2: Compare with (difference)n, which starts at the difference: first term − difference = the number to add$\mathstrut 40 - (-5) =$ $45$
   3. Step 3: So the nth term is (the number to add) + (difference)n$T_n = 45 - 5n$
3. Question 3Find a given term

   Find the 20th term of the sequence $-3, \ 0, \ 3, \ 6, \ \ldots$.

   Show the worked solution

   1. Step 1: Find the common difference: it is the number in front of n$\mathstrut 0 - (-3) =$ $3$
   2. Step 2: Compare with (difference)n, which starts at the difference: first term − difference = the number to add$\mathstrut (-3) - 3 =$ $-6$
   3. Step 3: Write the nth term$T_n = 3n - 6$
   4. Step 4: Put n = 20 into the nth term$3 \times 20 - 6 = 54$
4. Question 4Is it a term?

   Determine whether $107$ is a term of the sequence $5, \ 8, \ 11, \ 14, \ \ldots$.

   Show the worked solution

   1. Step 1: Find the common difference: it is the number in front of n$\mathstrut 8 - 5 =$ $3$
   2. Step 2: Compare with (difference)n, which starts at the difference: first term − difference = the number to add$\mathstrut 5 - 3 =$ $2$
   3. Step 3: Set the nth term equal to 107 and solve for n$3n + 2 = 107$$n = 35$
   4. Step 4: It is a term only if n is a whole number$n$ is a whole number, so $107$ is the 35th term: Yes.
5. Question 5Square numbers in the nth term

   Find an expression, in terms of $n$, for the $n$th term of the sequence $1, \ 7, \ 15, \ 25, \ \ldots$.

   Show the worked solution

   1. Step 1: The differences go up by 2 each time: compare with the square numbers 1, 4, 9, 16Term minus $n^{2}$: $0, 3, 6, 9$
   2. Step 2: What is left goes up by the same amount: find its nth term$3n - 3$
   3. Step 3: So the nth term is n² plus that$T_n = n^{2} + 3n - 3$
6. Question 6Sequences of fractions

   Find an expression, in terms of $n$, for the $n$th term of the sequence $\dfrac{4}{7}, \ \dfrac{6}{11}, \ \dfrac{8}{15}, \ \dfrac{10}{19}, \ \ldots$.

   Show the worked solution

   1. Step 1: The numerators go up by the same amount each time: find their nth term$2n + 2$
   2. Step 2: Do the same for the denominators$4n + 3$
   3. Step 3: So the nth term is$T_n = \dfrac{2n + 2}{4n + 3}$