<prompt> Acha S ionyeshe thamani ya jumlasum_n 19800 frac1sqrtn sqrtn2 - 1S inaweza kuonyeshwa kama p q sqrtr, ambapo p, q, na r ni nambari kamili chanya na r haiwezi kugawanywa kwa mraba wa msingi wowote. Amua p q r.  <response>: Ona kwamba sqrtn sqrtn2 - 1 frac1sqrt2sqrt2n 2sqrtn1n-1 frac1sqrt2leftsqrtn1sqrtn-1kulia. Hivyo, tunasum_n 19800 frac1sqrtn sqrtn2 - 1 sqrt2sum_n 19800 frac1sqrtn1sqrtn-1 frac1sqrt2sum_n 19800 kushotosqrtn1-sqrtn-1kuliaHuu ni mfululizo wa darubini; kumbuka kuwa tunapopanua majumuisho, masharti yote ya kati hughairi, na kutuacha na frac1sqrt2leftsqrt9801sqrt9800-sqrt1 -sqrt0kulia 70 49sqrt2, na pqr121. 