Mathematics in Ancient and Medieval India c8.s3.55

c8.s3.55

$t_n = (s2n. s)/((2²+2)(4²+4)...[(2n)²+2n]r2n) (n = 1,2,3...).$ The successive terms t₁, t₂, t₃...are, $t₁ = (s³)/(3!r²), t₂ = (s⁵)/(5!r⁴), t₃ = (s⁷)/(7!r⁶), t₄ = (s⁹)/(9!r⁸), ......$ Then according to the rule, $jīvā = (s - t₁) + (t₂ - t₃) + (t₄ - t₅) + ......$ $= s - (s³)/(3!r²) + (s⁵)/(5!r⁴) - (s⁷)/(7!r⁶) + (s⁹)/(9!r⁸) - (s¹¹)/(11!r¹⁰) + ...... (1)$ Again, $t'_n = (s2n. r)/((2²-2)(4²-4)...[(2n²)-2n]r2n) (n = 1,2,3,...)$ The successive tetms t'₁, t'₂, t'₃, ... are : $t'₁ = (s²)/(2!r), t₂ = (s⁴)/(4!r³), ... t'₆ = (s¹²)/(12!r¹¹)...$