৫+৫৫+৫৫৫+……………ধারাটির ১ম n সংখ্যক পদের যোগফল নির্নয় করুন?

ধরি, S = ৫+৫৫+৫৫৫+……………
S/5 = 1+11+111+………..
9S/5 = 9+99+999+…………
= (10-1)+(100-1)+(1000-1)+……………..
= (10+10²+10³+………..+10ⁿ) –(1+1+1+………..+n)
= 10(1+10+10²………….10ⁿ) – n
= 10{(10ⁿ-1)/(10-1)} – n
= {10(10ⁿ-1)/9}-n
S = 5/9 [{10(10ⁿ-1)/9} - n]
S = 50(10ⁿ-1)/81 – 5n/9

Shot Cut Rule:
a+aa+aaa+……………. n সংখ্যক পদের যোগফল
S = (a/9){(10/9)(10ⁿ-1) - n}
= (5/9){(10/9)(10ⁿ-1) - n}
= 50/81(10ⁿ-1) - 5n/9