Regroup a long sum of differences · 12 practice problems

Each bracket's value drops by $4$: $37, 33, 29, \ldots$, ending at $2-41=-39$. The values are evenly spaced, so the average is $\frac{37+(-39)}{2} = -1$, and $20$ brackets give $20 \times (-1) = -20$.

Another way: Pair the first bracket with the last: $(40-3)+(2-41) = 37 + (-39) = -2$, the second with the second-to-last gives $-2$ as well, and so on. With $20$ brackets there are exactly $10$ pairs, so the sum is $10 \times (-2) = -20$.

Each bracket's value drops by $4$: $47, 43, 39, \ldots$, ending at $2-51=-49$. The values are evenly spaced, so the average is $\frac{47+(-49)}{2} = -1$, and $25$ brackets give $25 \times (-1) = -25$.

Another way: Pair the first bracket with the last: $(50-3)+(2-51) = 47 + (-49) = -2$, the second with the second-to-last gives $-2$ as well, and so on. With $25$ brackets there are $12$ pairs and the middle bracket $(26-27)=-1$, so the sum is $12 \times (-2) + (-1) = -25$.

Each bracket's value drops by $4$: $61, 57, 53, \ldots$, ending at $2-65=-63$. The values are evenly spaced, so the average is $\frac{61+(-63)}{2} = -1$, and $32$ brackets give $32 \times (-1) = -32$.

Another way: Pair the first bracket with the last: $(64-3)+(2-65) = 61 + (-63) = -2$, the second with the second-to-last gives $-2$ as well, and so on. With $32$ brackets there are exactly $16$ pairs, so the sum is $16 \times (-2) = -32$.

Each bracket's value drops by $4$: $73, 69, 65, \ldots$, ending at $2-77=-75$. The values are evenly spaced, so the average is $\frac{73+(-75)}{2} = -1$, and $38$ brackets give $38 \times (-1) = -38$.

Another way: Pair the first bracket with the last: $(76-3)+(2-77) = 73 + (-75) = -2$, the second with the second-to-last gives $-2$ as well, and so on. With $38$ brackets there are exactly $19$ pairs, so the sum is $19 \times (-2) = -38$.

Each bracket's value drops by $4$: $85, 81, 77, \ldots$, ending at $2-89=-87$. The values are evenly spaced, so the average is $\frac{85+(-87)}{2} = -1$, and $44$ brackets give $44 \times (-1) = -44$.

Another way: Pair the first bracket with the last: $(88-3)+(2-89) = 85 + (-87) = -2$, the second with the second-to-last gives $-2$ as well, and so on. With $44$ brackets there are exactly $22$ pairs, so the sum is $22 \times (-2) = -44$.

Each bracket's value drops by $4$: $95, 91, 87, \ldots$, ending at $2-99=-97$. The values are evenly spaced, so the average is $\frac{95+(-97)}{2} = -1$, and $49$ brackets give $49 \times (-1) = -49$.

Another way: Pair the first bracket with the last: $(98-3)+(2-99) = 95 + (-97) = -2$, the second with the second-to-last gives $-2$ as well, and so on. With $49$ brackets there are $24$ pairs and the middle bracket $(50-51)=-1$, so the sum is $24 \times (-2) + (-1) = -49$.

Each bracket's value drops by $4$: $107, 103, 99, \ldots$, ending at $2-111=-109$. The values are evenly spaced, so the average is $\frac{107+(-109)}{2} = -1$, and $55$ brackets give $55 \times (-1) = -55$.

Another way: Pair the first bracket with the last: $(110-3)+(2-111) = 107 + (-109) = -2$, the second with the second-to-last gives $-2$ as well, and so on. With $55$ brackets there are $27$ pairs and the middle bracket $(56-57)=-1$, so the sum is $27 \times (-2) + (-1) = -55$.

Each bracket's value drops by $4$: $117, 113, 109, \ldots$, ending at $2-121=-119$. The values are evenly spaced, so the average is $\frac{117+(-119)}{2} = -1$, and $60$ brackets give $60 \times (-1) = -60$.

Another way: Pair the first bracket with the last: $(120-3)+(2-121) = 117 + (-119) = -2$, the second with the second-to-last gives $-2$ as well, and so on. With $60$ brackets there are exactly $30$ pairs, so the sum is $30 \times (-2) = -60$.

Each bracket's value drops by $4$: $127, 123, 119, \ldots$, ending at $2-131=-129$. The values are evenly spaced, so the average is $\frac{127+(-129)}{2} = -1$, and $65$ brackets give $65 \times (-1) = -65$.

Another way: Pair the first bracket with the last: $(130-3)+(2-131) = 127 + (-129) = -2$, the second with the second-to-last gives $-2$ as well, and so on. With $65$ brackets there are $32$ pairs and the middle bracket $(66-67)=-1$, so the sum is $32 \times (-2) + (-1) = -65$.

Each bracket's value drops by $4$: $147, 143, 139, \ldots$, ending at $2-151=-149$. The values are evenly spaced, so the average is $\frac{147+(-149)}{2} = -1$, and $75$ brackets give $75 \times (-1) = -75$.

Another way: Pair the first bracket with the last: $(150-3)+(2-151) = 147 + (-149) = -2$, the second with the second-to-last gives $-2$ as well, and so on. With $75$ brackets there are $37$ pairs and the middle bracket $(76-77)=-1$, so the sum is $37 \times (-2) + (-1) = -75$.

Each bracket's value drops by $4$: $169, 165, 161, \ldots$, ending at $2-173=-171$. The values are evenly spaced, so the average is $\frac{169+(-171)}{2} = -1$, and $86$ brackets give $86 \times (-1) = -86$.

Another way: Pair the first bracket with the last: $(172-3)+(2-173) = 169 + (-171) = -2$, the second with the second-to-last gives $-2$ as well, and so on. With $86$ brackets there are exactly $43$ pairs, so the sum is $43 \times (-2) = -86$.