<prompt> Kokotoa hadi sehemu mbili za desimali nambari sqrt12 sqrt12 sqrt1cdots2 sqrt12 sqrt1969 ambapo usemi una mizizi ya mraba 1969.  <response>: Let x_nsqrt12 sqrt12 sqrt1cdots2 sqrt12 sqrt1969 na usemi ulio na n alama za mizizi ya mraba. Kumbuka kwamba x_1-1sqrt2sqrt1969-1sqrt250 . Pia, kwa kuwa sqrt121sqrt21sqrt2, tuna beginaligned x_n1-1 sqrt2 sqrt12 x_n-sqrt121sqrt2frac2leftx_n- 1-sqrt2kuliasqrt12 x_nsqrt121sqrt2 frac x_n-1sqrt21sqrt2 . endaligned Kutoka hapa tunakisia kwamba x_1969-1sqrt2frac501sqrt21968 10-3, na kadirio la x_1969 lenye nafasi mbili za desimali linalingana na lile la 1sqrt22.41. Hoja hii inathibitisha pia kwamba kikomo cha mfuatano ni 1sqrt2. 