Cho c = 2 + 2^2 +2^3 + 2^4 + 2^5 + … + 2^100 a. Chứng minh c chia hết cho 31 b. Tìm số tự nhiên x biết 2^2x-1 – 2 = c

Question

Cho c = 2 + 2^2 +2^3 + 2^4 + 2^5 + … + 2^100
a. Chứng minh c chia hết cho 31
b. Tìm số tự nhiên x biết 2^2x-1 – 2 = c

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Hadley 4 tuần 2021-07-11T13:57:00+00:00 2 Answers 1 views 0

Answers ( )

    0
    2021-07-11T13:58:49+00:00

                                Bài giải

    a, Ta có : 

    c = 2+2$^{2}$ + 2$^{3}$+2$^{4}$ + 2$^{5}$ + … +2$^{100}$

    c = ( 2+2$^{2}$ + 2$^{3}$+2$^{4}$ + 2$^{5}$ ) + … + ( 2$^{96}$ + 2$^{97}$+2$^{98}$+ 2$^{99}$+2$^{100}$ )

    c = 2( 1+2 +2$^{2}$ + 2$^{3}$+2$^{4}$ ) + … + 2^96 ( 1+2 +2$^{2}$ + 2$^{3}$+2$^{4}$ )

    c = 2 . 31 + … + $2^{96}$   . 31 

    c = 31 ( 2 + … + $2^{96}$  ) chia hết cho 31

    0
    2021-07-11T13:58:50+00:00

    c = 2 + 2^2 +2^3 + 2^4 + 2^5 + … + 2^100

    =( 2 + 2^2 +2^3 + 2^4 + 2^5)+(2^6+2^7+2^8+2^9+2^10)+ … +(2^96+2^97+2^98+2^99+2^100)

    =2 ( 2 + 2^2 +2^3 + 2^4  )+2^6( 2 + 2^2 +2^3 + 2^4)+…+2^96( 2 + 2^2 +2^3 + 2^4)

    =2 ×31+2^6×31+…+2^96×31

    =31(2+2^6+…+2^96) CHIA HẾT CHO 31

    =>C chia hết cho 31

     

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