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Theorems on Integration

Which is larg...

Question

Which is larger : a) $(99^{50} + 100^{50})$ or b) ${(101)}^{50}$ .If u think a write 1 otherwise write 0 ?

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Solution

$(99)^{50} + (100)^{50}$
Now
$(101)^{50} - 100^{50} - 99^{50}$
$(1 + 100)^{50} - (1 - 100)^{50} - 100^{50}$
$= 2 [^{50} C_{1} 100 +^{50} C_{3} {.100}^{3} + . . .^{50} C_{49} 100^{49}] - 100^{50}$
$= 2 [^{50} C_{1} 100 +^{50} C_{3} {.100}^{3} + . . .^{50} C_{48} 100^{48}] +^{50} C_{49} 100^{49} +^{50} C_{49} 100^{49} - 100^{50}$
$= 2 [^{50} C_{1} 100 +^{50} C_{3} {.100}^{3} + . . .^{50} C_{48} 100^{48}] +^{50} C_{49} 100^{49} + 100^{49} [50 - 100]$
$= 2 [^{50} C_{1} 100 +^{50} C_{3} {.100}^{3} + . . .^{50} C_{48} 100^{48}] +^{50} C_{49} 100^{49} -^{50} C_{49} 100^{49}$
$= 2 [^{50} C_{1} 100 +^{50} C_{3} {.100}^{3} + . . .^{50} C_{48} 100^{48}] > 0$
Hence
$101^{50} > 99^{50} + 100^{50}$

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