1.

निम्न श्रेणी के अनन्त पदों का योग `(44)/(9)`है |`3+(3+d)(1)/(4)+(3+2d)(1)/(4^(2))`+.... तब d का मान ज्ञात कीजिए |

Answer» माना `" "S=3+(3+d)(1)/(4)+(3+2d)(1)/(4^(2))+...ooतक " "`....(i)
`rArr " "(1)/(4)S=3.(1)/(4)+(3+d)(1)/(4^(2))+(3+2d)(1)/(4^(3))+...ooतक ""`....(ii)
समीकरण (i) से समीकरण (ii) को घटाने पर
`(S-(1)/(4)S)=(3)/(4)S=3+(d)/(4)+(d)/(4^(2))+(d)/(4^(3))+...oo तक `
`=3+d[(1)/(4)+(1)/(4^(2))+(1)/(4^(3))+...oo तक]`
`3+d((1//4)/(1-1//4))=3+d((1//4)/(3//4))=3+(d)/(3)`
`:. " "(3)/(4)S=(d+9)/(3)" "rArr " "S=(4d+36)/(9)`
दिया है : `" "S=(44)/(9)`
`:. " "(44)/(9)=(4d+36)/(9)`
`rArr " "4d=44-36=8`
`rArr " "d=2`


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