What is the sum of all positive integers less than 100 which are multiples of five? (a) 950 (b) 1230 (c) 760 (d) 875 (Haryana Auction Recorder Exam, 2016)
Answer (1)
To find the sum of all positive integers less than 100 that are multiples of five, we can follow these steps:
Identify the Multiples of Five Less Than 100:
The multiples of five less than 100 are 5, 10, 15, ..., 95. This is an arithmetic sequence where the first term a = 5 and the common difference d = 5 .
Determine the Number of Terms in the Sequence:
To find the number of terms n in this sequence, we use the formula for the nth term of an arithmetic sequence:
nth term = a + ( n − 1 ) ⋅ d
So, if the last term is 95, we set up the equation:
95 = 5 + ( n − 1 ) ⋅ 5
Solving for n :
95 = 5 + 5 n − 5 95 = 5 n n = 19
Therefore, there are 19 terms in the sequence.
Calculate the Sum of the Sequence:
The sum of an arithmetic sequence can be found using the formula:
S n = 2 n ⋅ ( a + l )
where l is the last term of the sequence.
Substituting the known values:
S 19 = 2 19 ⋅ ( 5 + 95 ) S 19 = 2 19 ⋅ 100 S 19 = 950
Therefore, the sum of all positive integers less than 100 which are multiples of five is 950.
The Correct Multiple Choice Option:
The correct answer is option (a) 950.
