Consider two different arithmetic sequences

Question

Consider two different arithmetic sequences: 3, 14, 25, 36,47,58,69,80, successively by 11 and 2, 9, 16,23 successively by 7. What are the first 10 numbers which appear in both sequences

Answer 1

Define both sequences by:

3 + 11n

2 + 7m

For all integers n, m.

 

Set both sequences equal to each other to define when a number appears in both sequences:

3 + 11n = 2 + 7m

Rearranging the equation to see more easily when this holds true:

1 + 4n = 7*(m-n)

Define k = m - n.

1 + 4n = 7k.

 

When this holds true, 7k is every other odd number (i.e. 21 (7*3), 49 (7*7), 77 (7*11), etc)

1) k = 3

From this, we see from the equation, n = 5 and therefore, by definition of k, m = 8

Continuing for 10 k values:

k = 7 => n = 12, m = 19

k = 11 => n = 19, m = 30

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*Note that n increases by 7 and m increases by 11 each time, Thus to find the values in the original sequences, substitute the sequence:

n = 7*i - 2 (for integers i)

m = 11*i - 3

 

Thus, the values in the original sequence are when you substitute either n or m:

3 + 11*(7*i - 2) = 77*i -19

2 + 7*(11*i - 3) = 77*i - 19

*Note that both sequences have the same indexing, which is what we want.

 

The first 10 values in this set are when i = 1 to 10, which is adding 77 to each value after the first value of 58:

{58,135,212,289,366,443,520,597,674,751}

 

Answer 2

There's a trick to this one. Since the second sequence starts at a lower number, increase the number by 7, until you come to a number in the first sequence:

 

2, 9, 16, 23, 30, 37, 44, 51, [58]

 

58 is shared by both sequences. You need to find 9 more numbers like this one. When will the sequences share a number again? If you were counting up from 0, in both sequences:

 

0, 11, 22, 33, 44, 55, 66, [77]

0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, [77]

 

The first place you would meet is at the least common multiple (LCM) of 11 and 7, which is 77.

 

Therefore, in each of the sequences, you will meet at every 77 after 58.

 

58 "+ (77 * 0)", 58 + (77 * 1), 58 + (77 * 2), ..., 58 + (77 * 9)