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AP the sum of integers from 1 to 100 that are devisible by 2 or 5 is
Asked by thakuranurag0987 | 27 Dec, 2022, 02:31: PM
answered-by-expert Expert Answer
Sum S of arithmetic progression is given as
 
begin mathsize 14px style S space equals space n over 2 space left square bracket space 2 space a space plus space left parenthesis n minus 1 right parenthesis d space right square bracket end style
where  n is number of terms in series , a is first term and d is common differnce .
 
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Let S2 be sum of integers from 1 to 100 that are divisble by 2
 
S2 = 2 + 4 + 6 + ..................................................+98 + 100   .........................(1)
 
There are 50 terms in above series.
 
S2 = ( 50 /2 ) [ 4 + (49 × 2) ] = 2550
 
 
Let S5 be sum of integers from 1 to 100 that are divisble by 5
 
S5 = 5 + 10 + 15 + .....................95 + 100   ................................(2)
 
There are 20 terms in above series.
 
S5 = ( 20 /2 ) [ 10 + (19 × 5) ] = 1050
 
common terms in both the series given by eqn.(1) and (2) are
 
S10 = 10 + 20 + 30 ......................+90+100
 
There are 10 terms in above series
 
S10 = (10/2) [ 20 + 9 × 10 ]  = 550
 
Sum S of numbers from 1 to 100 that are divisible by 2 or 5 is
 
S = S2 + S5 - S10 =  2550 + 1050 - 550 = 3050
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