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If there be m AP’s beginning with unity whose common difference is 1, 2, 3 .... m . Show that the sum of their nth terms is (m/2) (mn -m+ n + 1).
Asked by rushabhjain.a | 21 Feb, 2019, 08:05: PM
Let tpn represent  n-th term of p-th arithmetic progression

we have, t1n = 1+(n-1)1
t2n = 1+(n-1)2
t3n = 1+(n-1)3
.....................
....................
tmn = 1+(n-1)m

Sum S of all n-th term is given by,   S = t1n + t2n + t3n +.................................+ tmn =  m +(n-1)[ 1+2+3................+m ]

S = m + (n-1) [ m (m+1)/2 ]  =  m +(m/2) (n-1)(m+1) = m + (m/2) [ nm -m +n -1] = (m/2)[nm -m +n +1]
Answered by Thiyagarajan K | 21 Feb, 2019, 11:20: PM

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