+4609 In each blank below a single digit is inserted such that the following six three-digit numbers, in this order, form an arithmetic sequence:
\( 1 \,\underline{\hspace{9pt}}\, \,\underline{\hspace{9pt}}\, \;\;,\;\; \,\underline{\hspace{9pt}}\, \,\underline{\hspace{9pt}}\, 9 \;\;,\;\; 2 \,\underline{\hspace{9pt}}\, 2 \;\;,\;\; \,\underline{\hspace{9pt}}\, 6 \,\underline{\hspace{9pt}}\, \;\;,\;\; 2 \,\underline{\hspace{9pt}}\, \,\underline{\hspace{9pt}}\, \;\;,\;\; \,\underline{\hspace{9pt}}\, 3 \,\underline{\hspace{9pt}} \)
What is the value of the next number in the sequence?
+128408 1 _ _ , _ _9, 2 _ 2 , _6_ , 2_ _ , _3_
The last number of the common difference must end in 3
And this must be a 2 digit number
So....the first number must end in 6
The the 4th number must be 265
So..... the 5th number must end in 8
And the last digit of the last term must end in 1 [ and begin with 3 ]
Putting all this together we have that
265 + 2d = 331
2d = 66 ⇒ d = 33
So the first term must be 265 - 3 (66) = 166
So....the series appears to be 166, 199, 232, 265, 298, 331
And the generating formula for the nth term is 166 + 33 (n - 1)
