________________
s = " [{ 21-12 d_ + ad } (n-1) + a ]
which is in its most general form. Another method of the same formula is given inl of the text.
Now comes the rule for finding the sum of the cubes of first n natural numbers which has been given to be equal to square of the sum of first n natural numbers.2
गच्छार्धवर्गराशीरूपाधिकगच्छवर्गसङ्गुणितः ।
धनसङ्कलितं प्रोक्तं गणितेऽस्मिन् गणिततत्त्वज्ञः ।। "The square of half of the number of terms is multiplied by the square of (the number of term increased by one) which gives rise to the sum of cubes of first n natural numbers as stated by mathematicians."
Algebraically. En = { n {n}" }
In the following stanza he has given a rule for finding out the sum of the cubes of the terms which are in A. P. This formula is in its most general form.
चित्यादिहतिर्मुखचयशेषघ्ना प्रचयनिघ्नचितिवर्गे।
आदौ प्रचयादूने वियुता युक्ताधिके तु धनचितिका ॥ Algebraically, if s = sum of terms in A. P.
a = first term, d = common difference n = number of terms
Sn = sum of the given series then Sn= Z[a + (n-1) d] = Sd + Sa (a-d)
where s = , [ 2a + (n-1) d]
or specifically (i) when a>d, Sn = + Sa (a-d) + Sad
(ii) when a<d, Sn - - Sa (a-d) + Sd
In the following stanza a rule has been given for finding out the sum of such a series whose each term is the sum of an A. P. of natural nos. having the number of terms equal to the term itself..
द्विगुणकोनपदोत्तरकृतिहतिरङ गाहृता चयायुता। आदिचयाहतियुक्ता व्येकपदघ्नादिगुणितेन ॥ सैकप्रभवेन युता षट्दलगुणितैव चितिचितिका ॥
168 168
299 301
1. Gss 6 2. Gss 6 3. Gs36 4. Gss 6
169
303
169
305-3051
आचार्यरत्न श्री देशभूषण जी महाराज अभिनन्दन पन्थ
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