To express the cubes of the given numbers as sums of consecutive odd numbers, we'll follow the appropriate mathematical method to break down each cube. The steps involve identifying the starting point and then generating the necessary sequence of odd numbers.
### Part a: [tex]\( 8^3 \)[/tex]
To express [tex]\( 8^3 \)[/tex] as the sum of consecutive odd numbers, we identify the starting odd number and then list the first 8 odd numbers starting from that point:
- Starting odd number for [tex]\( 8^3 \)[/tex]: 57
- Consecutive odd numbers: 57, 59, 61, 63, 65, 67, 69, 71
Thus, [tex]\( 8^3 \)[/tex] can be written as:
[tex]\[ 8^3 = 57 + 59 + 61 + 63 + 65 + 67 + 69 + 71 \][/tex]
### Part b: [tex]\( 12^3 \)[/tex]
For [tex]\( 12^3 \)[/tex], we again identify the first odd number in the sequence and continue for 12 terms:
- Starting odd number for [tex]\( 12^3 \)[/tex]: 133
- Consecutive odd numbers: 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155
Thus, [tex]\( 12^3 \)[/tex] can be written as:
[tex]\[ 12^3 = 133 + 135 + 137 + 139 + 141 + 143 + 145 + 147 + 149 + 151 + 153 + 155 \][/tex]
### Part c: [tex]\( 15^3 \)[/tex]
For [tex]\( 15^3 \)[/tex], identify the sequence starting point and enumerate 15 terms:
- Starting odd number for [tex]\( 15^3 \)[/tex]: 211
- Consecutive odd numbers: 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 233, 235, 237, 239
Thus, [tex]\( 15^3 \)[/tex] can be written as:
[tex]\[ 15^3 = 211 + 213 + 215 + 217 + 219 + 221 + 223 + 225 + 227 + 229 + 231 + 233 + 235 + 237 + 239 \][/tex]
### Part d: [tex]\( 17^3 \)[/tex]
Next, for [tex]\( 17^3 \)[/tex], list the sequence beginning at the identified odd number and proceed for 17 terms:
- Starting odd number for [tex]\( 17^3 \)[/tex]: 273
- Consecutive odd numbers: 273, 275, 277, 279, 281, 283, 285, 287, 289, 291, 293, 295, 297, 299, 301, 303, 305
Thus, [tex]\( 17^3 \)[/tex] can be written as:
[tex]\[ 17^3 = 273 + 275 + 277 + 279 + 281 + 283 + 285 + 287 + 289 + 291 + 293 + 295 + 297 + 299 + 301 + 303 + 305 \][/tex]
### Part e: [tex]\( 19^3 \)[/tex]
Finally, for [tex]\( 19^3 \)[/tex], establish the sequence starting number and continue for 19 terms:
- Starting odd number for [tex]\( 19^3 \)[/tex]: 343
- Consecutive odd numbers: 343, 345, 347, 349, 351, 353, 355, 357, 359, 361, 363, 365, 367, 369, 371, 373, 375, 377, 379
Thus, [tex]\( 19^3 \)[/tex] can be written as:
[tex]\[ 19^3 = 343 + 345 + 347 + 349 + 351 + 353 + 355 + 357 + 359 + 361 + 363 + 365 + 367 + 369 + 371 + 373 + 375 + 377 + 379 \][/tex]
By following the appropriate steps for each cube, we have successfully expressed these cubes as sums of consecutive odd numbers.
