We know that power is current squared times resistance, and also voltage squared divided by resistance.P=(I^2)*Ri.e. P is directly proportional to R..................(1)P=(V^2)/Ri.e. P is inversely proportional to R................(2)This presents a contradiction. Why does this contradiction occur and which among (1) and (2) is correct?I've been wondering about this for long and am confused. Help will be greatly appreciated! Thank you.

Current in the circuit is not independent..
It depends on resistance.
So u cannot take I as independent quantity
Since on changing the resistance both i and P will change.
To find relationship u need to have constant before proportionality
like:
P is proportional to (contant* R)
So we have to take
P = V^2/R
Since V is constant in the circuit
and thus we can say
P is inversely proprtional to R^2
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P = I^2R is used when current flowing in every component of the circuit is constant. This is the case of series combination of the devices in the circuit. P = V^2/R is used when potential difference (V) across every component of the circuit is constant. This is the case of parallel combination of the devices in the circuit.

Approved

P=(I^2)*R

IF CURRRENT IS CONSTANT THEN POWER IS DIRECTLY PROPORTIONAL TO RESISTANCE

AS HAPPENS IN SERIES CONNECTION.SUPPOSE THREE RESISTANCE 2,3&5OHM ARE CONNECTED IN SERIES THEN POWER CONSUMPTION

ORDER IS 5ohm>3ohm>2ohm.THIS IS BECAUSE DUE TO CONSTANT CURRENT IN THREE RESISTANCES" POWER CONSUMED~RESSTANE"

#P=V^2/R

IF POTENTIAL DIFF. IS CONSTANT THEN POWER IS INVERSELY PROPORTIONAL TO RESISTANCE

AS HAPPENS IN PARALLEL COMBINATION OF RESISTANCES. LET THREE RESISTOR OF INDIV. RESISTANCES 2,3&5 ohm CONNECTED IN PARALLEL THEN THE ORDER OF POWER CONSUMPTION IS AS FOLLOW

2ohm>3ohm>5ohm

the above trend is because power varies inversly with resistance in parallel combination

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this relation actually depends on the constant value if the current throughout the circuit is constant( as in case of series combination), resistance is directly proportional to power that is P=I^2* R if the voltage is constant throughout the circuit (as in case of parallel combination), resistance is inversely proportional to power that is P=V^2/R