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Question

Prove that:
(i) 13+7+17+5+15+3+13+1=1
(ii) 11+2+12+3+13+4+14+5+15+6+16+7+17+8+18+9=2


Solution

(i)

To prove: 13+7+17+5+15+3+13+1=1

Lets take LHS and then equate it to RHS.

LHS =13+7+17+5+15+3+13+1

Lets do rationalize the denominator of each term,

=[13+7×3737]+[17+5×7575]+[15+3×5353]+[13+1×3131]

=3732(7)2+75(7)2(5)2+53(5)2(3)2+31(3)212

=3797+7575+5353+3131

=372+752+532+312

=37+75+53+312

=22

=1

= RHS

LHS = RHS

Hence, proved.

(ii) 

To prove:

11+2+12+3+13+4+14+5+15+6+16+7+17+8+18+9=2

Lets take LHS and then equate it to RHS.

LHS =11+2+12+3+13+4+14+5+15+6+16+7+17+8+18+9

Lets do rationalize the denominator of each term,

=[11+2×2121]+[12+3×3232]+[14+3×4343]+[14+5×5454]+[15+6×6565]+[16+7×7676]+[17+8×8787]+[18+9×9898]

=21(2)212+32(3)2(2)2+43(4)2(3)2+54(5)2(4)2+65(6)2(5)2+76(7)2(6)2+87(8)2(7)2+98(9)2(8)2

=2121+3232+4343+5454+6565+7676+8787+9898

=211+321+431+541+651+761+871+981

=21+32+43+54+65+76+87+98

=91

=31

=2

LHS = RHS

So, 11+2+12+3+13+4+14+5+15+6+16+7+17+8+18+9=2

Hence, proved.


Mathematics
RS Aggarwal (2020, 2021)
All

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