Arithmetic Sequence problem involving terms of the sequence and the value of that term

Question

The full question is here:
In an arithmetic sequence, the first term is $2$ and the second term is $5$. Term number $N$ of the sequence has a value of $M$, such that $M$ is the largest two-digit number in the sequence. What is the value of $M + N$?

I tried to make a possible sequence and solve that problem from there but have no success, in fact I kind of don't understand where to start. Any help will be appreciated, thank you.

Answer 1

From the first two elements of the sequence, the arithmetic sequence must be of the form $(x_n)$ with $x_n = 3n-1$.

The largest 2-digit number in this sequence is $3 \cdot 33-1=98$, and so $M+N = 98+33 = 131$.

Answer 2

Hint: Given that $$a_1=2,a_2=5$$ so we get $$5=2+d$$ so $$d=3$$ and you will get $$a_N=2+(N-1)\cdot 3=M$$

Answer 3

So, $100>2+(N-1)(5-2)\iff3N<100-2+3$

$\implies N<101/3<34$

Answer 4

Your sequence is the following : $$\forall n \in \mathbb{N}^*, \quad u_n = 2 + 3(n-1)$$

It consists of all the numbers that are equal to $2$ modulo $3$ : in the $90$-numbers, you have $92$, $95$, and $98$.

So the biggest is $M= 98$, and you have $u_N = M$ if and only if $$2 + 3(N-1)= 98, \quad i.e. \quad N=33$$

So $$N+M = 131$$

Answer 5

HINT:

In an arithmetic sequence you always have

$$ a_{n+1}=a_n+d $$

so you can figure out what $d$ is.

Furthermore

$$ a_{M}=a_1+(M-1)\cdot d $$

Hope I could help