Math

QuestionFind the number of marbles in the sixth line if each line follows the rule: nk=2nk11n_{k} = 2n_{k-1} - 1, starting with n1=2n_1 = 2.

Studdy Solution

STEP 1

Assumptions1. The first line has marbles. . Each consecutive line has a number of marbles that is one less than twice the number of marbles in the previous line.
3. We need to find the number of marbles in the sixth line.

STEP 2

We can express the rule as a recursive formula, where the number of marbles in the nth line (M_n) is one less than twice the number of marbles in the (n-1)th line (M_{n-1}).
Mn=2Mn11M_n =2M_{n-1} -1

STEP 3

We know that the first line (n=1) has2 marbles, so we can use this as our base case.
M1=2M1 =2

STEP 4

We can now use the recursive formula to find the number of marbles in the second line (n=2).
M2=2M11M2 =2M1 -1

STEP 5

Substitute the value of M1M1 into the formula to calculate M2M2.
M2=2(2)1M2 =2(2) -1

STEP 6

Calculate the number of marbles in the second line.
M2=41=3M2 =4 -1 =3

STEP 7

Repeat the process for the third line (n=3).
M3=2M21M3 =2M2 -1

STEP 8

Substitute the value of M2M2 into the formula to calculate M3M3.
M3=2(3)1M3 =2(3) -1

STEP 9

Calculate the number of marbles in the third line.
M3=6=5M3 =6 - =5

STEP 10

Repeat the process for the fourth line (n=4).
M4=2M3M4 =2M3 -

STEP 11

Substitute the value of M3M3 into the formula to calculate M4M4.
M4=(5)M4 =(5) -

STEP 12

Calculate the number of marbles in the fourth line.
M4=10=9M4 =10 - =9

STEP 13

Repeat the process for the fifth line (n=5).
M5=2MM5 =2M -

STEP 14

Substitute the value of M4M4 into the formula to calculate MM.
M=2(9)M =2(9) -

STEP 15

Calculate the number of marbles in the fifth line.
M5=18=17M5 =18 - =17

STEP 16

Finally, repeat the process for the sixth line (n=6).
M6=2M5M6 =2M5 -

STEP 17

Substitute the value of M5M5 into the formula to calculate M6M6.
M6=2(17)M6 =2(17) -

STEP 18

Calculate the number of marbles in the sixth line.
M6=34=33M6 =34 - =33So, the sixth line must have33 marbles. The answer is C.33.

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