What's the solution to this puzzle?

Question

I saw this on Instagram with no solution and was wondering what the answer is. I got $33$.

$$1+4=5$$

$$2+5=12$$

$$3+6=21$$

$$8+11=?$$

Answer 1

$a_n + b_n = c_n$

should in fact be interpreted as

$a_n \cdot b_n + n = c_n$

with

$b_n = a_n + 3$

and the following recursive relation for $a_n$

$a_n = a_{n-1} \cdot a_{n-2} + (n-2),\:\:$ $n \geq 3$

$a_0 = 0, \:\:\:\:a_1 = 1, \:\:\:\:a_2 =2$

---

$a_1 \cdot b_1 + 1 = c_1$

$a_{2} \cdot b_{2} + 2 = c_{2}$

$a_{3} \cdot b_{3} + 3 = c_{3}$

$a_{4} \cdot b_{4} + 4 = c_{4}$

give us

$1+4=5$

$2+5=12$

$3+6=21$

$8+11=c_4$

and

$c_{4}$

$= a_{4} \cdot b_{4} + 4 = 92$

or to demonstrate the recursion

$c_{4}$

$= (a_{3} \cdot a_{2} + 2) \cdot b_{4} + 4 $

$= (3 \cdot 2 + 2) \cdot 11 + 4$

$= 8 \cdot 11 + 4 = 92$

---

This results in the following sequence

$1+4=5$

$2+5=12$

$3+6=21$

$8+11=92$

$200 + 203 = 40605$

$4824 + 4827 = 23285454$

$964808 + 964811 = 930857371296$

$\vdots$ $\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\vdots$

---

Im pretty sure this is what they had in mind :)

Answer 2

If you number your ‘equations’ as $a_n+b_n =c_n$, you have the relation $c_n=a_n+b_n+c_{n-1}$. Hence \begin{align*} 4+7&=32,\\5+8&=45,\\6+9&=60,\\7+10&=77,\\8+11&=96.\end{align*}

Answer 3

Just for kicks, define \begin{equation} "a+b=c"\quad\Longleftrightarrow\quad a=\left\lfloor \frac{c}{b} \right\rfloor \end{equation} and I propose $88$ as a result :)

Answer 4

A possible law is $$a+b=ab+a$$ (this is valid for the three first sums) so an answer is $$8+11=8\cdot11+8=96$$

It could be possible maybe another law satisfying the three first sums.

Answer 5

I got 40...

But I guess there could be any number, because you can use the Lagrange Interpolation Polynomial

Answer 6

96 For each line add the two numbers and store it in D. Add D to the C from the previous step.

A+B=C     D
1+4=5     5   (4+1 gives 5)       
2+5=12    7   (5+2 is 7. Add 7 to 5 = 12 from previous step)
3+6=21    9   (6+3 is 9. Add 9 to 12 = 21 from previous step)
4+7=32   11   (4+7 is 11. Add 11 to 21 = 32 from previous step)
5+8=45   13 
6+9=60   15 
7+10=77  17
8+11=96  19