The general term

Question

The general term
What is the general term of
7,13,25,49,97

Answer 1

general term
3*2^n+1
1st term=3*2+1=7
2nd term=3*4+1=13
3rd term =3*8+1=25
4th term=3*16+1=49
5th term=3*32+1=97

Answer 2

請環保一下，不要製造知識層的污染！

Answer 3

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Answer 4

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Answer 5

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Answer 6

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Answer 7

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Answer 8

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Answer 9

T(1) = 7

T(2) = 13 = T(1) + 6 = 7 + 6

T(3) = 25 = T(2) + 12 = 7 + 6 + 2(6) = 7 + (1 + 2)(6) = 7 + (2^0 + 2^1)(6)

T(4) = 49 = T(3) + 24 = 7 + (1 + 2)(6) + 4(6) = 7 + (1 + 2 + 4)(6) = 7 + (2^0 + 2^1 + 2^2)(6)

T(5) = 97 = T(4) + 48 = 7 + (1 + 2 + 4)(6) + 8(6) = 7 + (1 + 2 + 4 + 8)(6) = 7 + (2^0 + 2^1 + 2^2 + 2^3)(6)

....

T(n) = 7 + [2^0 + 2^1 + 2^2 + ... + 2^(n-2)](6)

Consider x = 2^0 + 2^1 + 2^2 + ... + 2^(n-2),

x = 2^0 + 2^1 + 2^2 + ... + 2^(n-2) ..... (1)

2x = 2^1 + 2^2 + ... + 2^(n-2) + 2^(n-1) ..... (2)

(2) - (1),

x = 2^(n-1) - 2^0 = 2^(n-1) - 1


Therefore, T(n) = 7 + [2^(n-1) - 1](6) = 7 + 6[2^(n-1) - 1].

2007-02-18 21:06:14 補充：
不好意思, 太大意了, 還沒把答案約簡, Dave Kwok ~ Totti的答案是對的.T(n)= 7 6[2^(n-1) - 1]= 7 6 * 2^(n-1) - 6= 3 * 2 * 2^(n-1) 1= 3 * 2^n 1

2007-02-18 21:07:15 補充：
不好意思, 太大意了, 還沒把答案約簡, Dave Kwok ~ Totti的答案是對的.T(n)= 7 十 6[2^(n-1) - 1]= 7 十 6 * 2^(n-1) - 6= 3 * 2 * 2^(n-1) 十 1= 3 * 2^n 十 1

Answer 10

T(2)-T(1)=13-7=6
T(3)-T(2)=25-13=12
T(4)-T(3)=24
Which implies T(n+1)-T(n)=6(2)^(n-1)
T(2)=T(1)+6
T(3)=T(1)+6+6(2)
......
T(n)=T(1)+6+6(2)+......+6(2)^(n-1)
=T(1)+6[2^(n-1)-1]
=7+6[2^(n-1)-1]

2007-02-19 13:19:52 補充：
實際上7+6[2^(n-1)-1]也不是不對......