Question

The tens place digit of 1! + 2! + ------- + 29! :

Answer 1

Hint: Expand the first ten factorials of the series and you will find after 4! The unit digits of the remaining factorials are 0 so add the unit digit of 1! , 2!, 3!, 4!, you will find a double digit number from the addition of unit digits then keep the tens place number from this double digit and add in the number that comes from the addition of tens place digit from 4!to9! From 10! Onwards, the tens place value of digit is 0 so adding the tens value of these numbers won’t bring any change.

Complete step-by-step answer:

First, we expand the first 10 factorials starting from 1:
1 + 2 + 6 + 24 + 120 + 720 + 5040 + 40320 + 362880 + 3628800
Now, we can see the first 4 factorials, 1! 2! 3! 4! are ending with a non-zero unit digit so we are adding the unit digit of these numbers. In the given below, I am adding the underlined numbers:
1 + 2 + 6 + 24; the unit place digits addition (1 + 2 + 6 + 4) gives 13………………(1)
Now, from 4! To 9! you can see the tens place digits are non-zero,
24 + 120 + 720 + 5040 + 40320 + 362880
Now, adding the tens place digits of these numbers (here, I have shown the tens place digit by underline) we get:
2 + 2 + 2 + 4 + 2 + 8 = 20
Now, from the expression (1) we get 13 so add the tens digit of number 13 in the above result 20. Here, we are adding tens digit of 13 in 20 because the question demands tens digit of the factorial series summation. Adding 1 and 20 will give 21 so the tens digit is 1.
Hence, the final answer of the question is (1).

Note: You may be tempted to think like the series have some general expression for summation like A.P., G.P., there must be some general expression for the summation of factorial series. The answer is yes, there is an expression but it is not in the syllabus so you will learn in the higher class.