, 2020

The number of letters in each row after the first is twice the number of letters in the previous row.

Since Row 4 has 8 letters, then Row 5 has $2\times8=16$ letters, and Row 6 has $2\times16=32$ letters. Alternatively, we can continue the pattern to Row 6 as shown. Row 1 $A$ Row 2 $BB$ Row 3 $A$$A$$A$$A$ Row 4 $BBBBBBBB$ Row 5 $A$$A$$A$$A$$A$$A$$A$$A$$A$$A$$A$$A$$A$$A$$A$$A$ Row 6 $BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB$ If the pattern consists of 6 rows, the total number of letters is $1+2+4+8+16+32=63$. Solution 1 If the total number of letters in the pattern is 63, then there are 6 rows in the pattern (as we saw in part (b)). Counting, we get that there are $1+4+16=21$ $A$’s, and $2+8+32=42$ $B$’s. Solution 2 Notice that in Row 2 there are twice as many $B$’s as there are $A$’s in Row 1, and in Row 4 there are twice as many $B$’s as there are $A$’s in Row 3. Further, the rows alternate between $A$’s and $B$’s and the number of letters in each row is twice the number of letters in the previous row, and so this pattern continues. Thus, if there are an even number of rows in the pattern, then the total number of $B$’s in the pattern is twice the total number of $A$’s, and so in this case $\frac13$ of the letters in the pattern are $A$’s and $\frac23$ of the letters are $B$’s. If the total number of letters in the pattern is 63, then there are 6 rows in the pattern (as we saw in part (b)), and so the number of $A$’s in the pattern is $\frac13\times63=21$ and the number of $B$’s is $\frac23\times63=2\times21=42$. Solution 1 We begin by determining the number of rows in the pattern given that the total number of letters is 4095. We may do this by counting the number of $A$’s and $B$’s in each row and keeping a running total of the number of letters in the pattern after each complete row. Row Number 1 2 3 4 5 6 7 8 9 10 11 12 Number of $A$’s 1 0 4 0 16 0 64 0 256 0 1024 0 Number of $B$’s 0 2 0 8 0 32 0 128 0 512 0 2048 Number of Letters 1 3 7 15 31 63 127 255 511 1023 2047 4095 If the pattern has 12 complete rows, there are a total of 4095 letters, of which $1+4+16+64+256+1024=1365$ are $A$’s and $2+8+32+128+512+2048=2730$ are $B$’s. Thus, if there are 4095 letters in the pattern, the difference between the number of $A$’s and the number of $B$’s is $2730-1365=1365$. Solution 2 We begin by determining the number of rows in the pattern given that the total number of letters is 4095. Since $4095=1+2+4+8+16+32+64+128+256+512+1024+2048$ and the sum on the right side of this equation has 12 terms, then a pattern with 4095 letters contains exactly 12 complete rows. Since 12 is an even number of rows, we may use the result from Solution 2 in part (c) to determine that the pattern has $\frac13\times4095=1365$ $A$’s and $\frac23\times4095=2\times1365=2730$ $B$’s. Thus, if there are 4095 letters in the pattern, the difference between the number of $A$’s and the number of $B$’s is $2730-1365=1365$. (Alternatively, we may have concluded that if $\frac23$ of the letters are $B$’s and $\frac13$ are $A$’s, then the difference between the number of $A$’s and $B$’s is $\frac23-\frac13=\frac13$ of the total number of letters, or $\frac13\times4095=1365$.)