Question

1 , 3 , 6 , 10 , 15

1) What is the nth term of the sequence?

2) Use the nth term to work out the 21st term of the sequence.

Answer 1

First difference,

`\Delta_(1) = a_(2) - a_(1) = 3 - 1 = 2 = 1 + 1`

Second difference,

`\Delta_(2) = a_(3) - a_(2) = 6 - 3 = 3 = 2 + 1`

Third difference,

`\Delta_(3) = a_(4) - a_(3) = 10 - 6 = 4 = 3 + 1`

And so on.

Assuming the pattern holds on, we see that

i-th difference,

`\Delta_(i) = a_(i + 1) - a_(i) = i + 1`

`\implies a_(i + 1) = a_(i) + i + 1`

Then, nth term is,

`\implies a_(n) = a_(n - 1) + n`

`= a_(n - 2)+ (n + (n - 1))`

`= a_(n - 3) + (n + (n - 1) +(n - 2))`

`= a_(n - (n - 1)) + \sum \limits^(n - 2)_(k = 0)(n - k)`

`= a_1 + \sum \limits^(n - 2)_(k = 0)n- \sum \limits^(n - 2)_(k = 0)k`

`= a_1 +n(n -2 + 1 )- (1)/(2) (n - 2)(n - 1)`

`= a_1 +n(n -1 )- (1)/(2) (n - 2)(n - 1)`

`= a_1 +(n -1 )(n- (1)/(2) (n - 2))`

`= a_1 + (1)/(2) (n -1 )(2n- n + 2)`

`= a_1 + (1)/(2) (n -1 )(n + 2)`

`\implies a_(n) = 1 + (1)/(2) (n -1 )(n + 2)`

Now, the 21st term in the sequence is,

`\implies a_(21) = 1 + (1)/(2) (21 -1 )(21 + 2)`

`= 1 + (1)/(2) * 20 * 23`

`= 231`