HELPPp me asap please I'm hopeless The arithmetic progressions and have some common values. What is the largest value less than 500 that they have in common?

Work Shown:

We have two sequences, which we'll call A and B

Both sequences are arithmetic. The first sequence starts at 2, and increases by 3 each time. Sequence B starts at 3 and increases by 7 each time.

The terms in bold are what the two sequences have in common.

It turns out that the bold terms follow an arithmetic sequence of their own.

The sequence {17, 38, 59, ...} is arithmetic.

Call this sequence C. This sequence starts at 17 and increases by 21 each time. The nth term for sequence C is

C(n) = a+d(n-1)

C(n) = 17+21(n-1)

C(n) = 17+21n-21

C(n) = 21n-4

Plugging in n = 1 leads to C(n) = 17. Also, plug in n = 2 and you should get C(n) = 38. Plugging in n = 3 leads to C(n) = 59. And so on.

The question is now "what is the largest term in sequence C such that it is less than 500?".

In other words, we want to find the term C(n) where C(n) < 500 and C(n) is as large as possible.

Use a bit of algebra to get

C(n) < 500

21n - 4 < 500

21n < 500+4

21n < 504

n < 504/21

n < 24

So n must be smaller than 24 to get what we want.

Because n is a natural number (positive whole number), we drop to n = 23 to find that

C(n) = 21n-4

C(23) = 21(23)-4

C(23) = 479

The largest value, less than 500, the two original sequences have in common is 479.

If we didn't have the restriction "less than 500", then there would be no largest value as sequence C goes on forever.