Find the next two terms of the sequence: \( 5, 8, 11, 14, \ldots \)
Common difference \( 3 \). Next: \( 17, 20 \).
Sequences
8 exam-style questions on arithmetic sequences, nth term, geometric sequences and special sequences.
3 Easy 3 Medium 2 Hard
Find the \( n \)th term of the sequence \( 3, 7, 11, 15, \ldots \)
\( 4n - 1 \).
Is \( 47 \) in the sequence \( 5, 9, 13, 17, \ldots \)? Explain.
nth term \( = 4n + 1 \). \( 4n + 1 = 47 \) ⇒ \( n = 11.5 \), not integer, so no.
Find the \( n \)th term of \( 2, 8, 18, 32, \ldots \) (Hint: compare to \( n^2 \)).
Terms are \( 2n^2 \). Check: \( 2, 8, 18, 32 \) — yes \( 2n^2 \).
The \( n \)th term of a sequence is \( 5n - 2 \). Find the 20th term and the position of the term equal to \( 73 \).
20th: \( 5(20)-2 = 98 \). \( 5n-2=73 \) ⇒ \( n=15 \).
A geometric sequence has first term \( 5 \) and common ratio \( 2 \). Find the 6th term.
\( 5 \times 2^5 = 5 \times 32 = 160 \).
The \( n \)th term of a quadratic sequence is \( an^2 + bn + c \). The first three terms are \( 4, 9, 18 \). Find \( a \), \( b \) and \( c \).
\( n=1 \): \( a+b+c=4 \). \( n=2 \): \( 4a+2b+c=9 \). \( n=3 \): \( 9a+3b+c=18 \). Diff: \( 3a+b=5 \), \( 5a+b=9 \) ⇒ \( 2a=4 \) ⇒ \( a=2 \), \( b=-1 \), \( c=3 \). nth term \( = 2n^2 - n + 3 \).
An arithmetic sequence has first term \( a \) and common difference \( d \). The 4th term is \( 17 \) and the 9th term is \( 32 \). Find \( a \) and \( d \).
\( a + 3d = 17 \), \( a + 8d = 32 \). Subtract: \( 5d = 15 \) ⇒ \( d = 3 \). \( a + 9 = 17 \) ⇒ \( a = 8 \).