This question was uploaded on 16/08/21 on social media accounts.
Given four variables such that sum of first three is equal to fourth, sum of reciprocals of first three is equal to reciprocal of fourth. Sum of cubes of first three is 8. Find sum of all variables.
We have to find,
p + q + r + s = ?
But,
p + q + r = s
is given,
⇒ p + q + r + s = 2s
So we have to find value of 's'
Step 1:
We know that,
\[p^3+q^3+r^3-3pqr=(p+q+r)(p^2+q^2+r^2-pq-qr-rp)\]
Step 2:
Given,
\[\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{s}\]
\[\Rightarrow pq+qr+rp=\frac{pqr}{s}\]
Step 3:
Given
p + q + r = s
And also we know that,
\[p^2+q^2+r^2=(p+q+r)^2-2(pq+qr+rp)\]
Now,
\[\Rightarrow8-3pqr=(s)\left((p+q+r)^2-3(pq+qr+rp)\right)\]
\[\Rightarrow 8-3pqr=(s)\left(s^2-3\frac{pqr}{s}\right)\]
\[\Rightarrow 8-3pqr=s^3-3pqr\]
\[\Rightarrow s^3=8\]
\[\Rightarrow s=2\]
Step 4:
We got value of s:
Now,
p + q + r + s = 2s = 4
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