I should know this by now...

Problem #7

There is a set of 160 numbers, beginning at 6, with each subsequent term increasing by an increment of 3. What is the average of this set of numbers?

I don't understand the equation for solving the 160th term.

Why is the formula 6 + (159 X 3) ?

To determine the formula, let's try with easier examples first.

Example with 2 numbers:
6, 9

Formula to get 9:
9 = 6 + 3
9 = 6 + (1*3)

The '1' term in the parenthesis above is actually 1 less than the number of terms (2-1=1). You will see why this is important in a second.

Thus,
9 = 6 + [(2-1)*3]



Let's try another example.

Example with 3 numbers:
6, 9, 12

Formula to get 12:
12 = 6 + 3 + 3
12 = 6 + (2*3)

The '2' term in parenthesis above is also 1 less than the number of terms (3-1=2).

Thus,
9 = 6 + [(3-1)*3]

Seems like a pattern is forming.



If you're still unsure, try another example.

Example with 4 numbers:
6, 9, 12, 15

Formula to get 15:
15 = 6 + 3 + 3 + 3
15 = 6 + (3*3)

The '3' term in parenthesis is again 1 less than the number of terms (4-1=3).

Thus,
9 = 6 + [(4-1)*3]

I think the pattern is obvious now.



Try to generalize the formula.

Example with n numbers:
nth number = 6 + [(n-1)*3]


Hence with 160 numbers:
160th number = 6 + [(160-1)*3]
160th number = 6 + (159*3)


Sorry for the lengthy explanation. Hope it helps.

Can this also be figured another way?

If the rule for the sequence is determined to be kn + x, the first term in the sequence having the value of 6:

First term --> n = 1

3n + x = 6

3(1) + x = 6

x = 3

160th term:

3(160) + 3 = 483

For average:

6 + 483
2

= 489 / 2

= 244.5

The key is to recognize that the language of the problem tells us that this is an arithmetic sequence. (increases by 3 each time = arithmetic)

tmmyc lays this out nicely. The standard formula for an arithmetic sequence is:
a-sub-n = a-sub-1 + (n-1)d

Where a-sub-1 = the first term in the sequence
n = the number of terms
d = the difference between each term
a-sub-n = the desired term of the sequence

Should memorize the above for the test if you are looking for a 700+ score.

To the second guest, how did you determine that the rule is kn+x? Are you using a variant on the arithmetic sequence formula?