For the given sequence, find the next term.
1, 2, 4, 7, 11, ..........

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Solution

Let $a_{1} = 1$
We can define the sequence recursively as
$a_{n + 1} = a_{n} + n$
$∴ a_{2} = a_{1} + 1 = 1 + 1 = 2$
$a_{3} = a_{2} + 2 = 2 + 2 = 4$
$a_{4} = a_{3} + 3 = 4 + 3 = 7$
$a_{5} = a_{4} + 4 = 7 + 4 = 11$
$a_{6} = a_{5} + 5 = 11 + 5 = 16$
$a_{7} = a_{6} + 6 = 16 + 6 = 22$
$a_{8} = a_{7} + 7 = 22 + 7 = 29$
$a_{9} = a_{8} + 8 = 29 + 8 = 37$
Therefore, the next four terms are
$16, 22, 29, 37.$

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