2026/084) What is the next number in the sequence 9, 12, 17, 24, 33?

Let us loom at the 1st order differences they are $3,5,7,9$

Method 1:(Short form)

next difference is 11 and next number is 33 + 11 = 44

Method 2(Finding Polynomial form of the number) 

Now 2nd order difference are $2$ 

So the terms are defined by a quadratic polynomial and

$T_n= an^2 + bn+c$ for some a,b ,c 

Putting n =1 ,2, 3 respectively we get

$a+b+c = 9\cdots(1)$

$4a+ 2b + c = 12\cdots(2)$

$9a + 3b+c = 17\cdots(3)$ 

 Subtract (1) from(2) to get

$3a + b = 3\cdots(4)$

Subtract (2) from(3) to get

$5a + b = 5\cdots(5)$

Subtracting (4) frpm (5) we get $2a = 2$ or $a=1$

Putting value of a in (4) we get $b=0$

Putting value of a and b in (1) we get c = 8 

So $n^{th}$ term is $n^2+8$

 Next  number is $6^2+8=44$