Show that $\left(x^{2}+x y+y^{2}\right),\left(z^{2}+x z+x^{2}\right)$ and $\left(y^{2}+y z+z^{2}\right)$ are consecutive terms of an A.P., if $x, y$ and $z$ are in A.P.

Step 1: Understand the condition for A.P.

For three terms to be in an Arithmetic Progression (A.P.), the middle term must be the average of the other two terms. This means that the difference between the second and first term is equal to the difference between the third and second term. So, if $x, y, z$ are in A.P., then $y - x = z - y$, which simplifies to $2y = x+z$.

Step 2: Define the terms to be checked

We are given three expressions and need to show that they form an A.P. Let's label these expressions as $A$, $B$, and $C$ for clarity. We need to prove that $2B = A+C$.

Step 3: Calculate $A+C$

First, we calculate the sum of the first and third terms, $A+C$. We combine like terms to simplify the expression.

Step 4: Substitute $2y = x+z$ into $A+C$

From the condition that $x, y, z$ are in A.P., we know $x+z = 2y$. We substitute this into the expression for $A+C$ to simplify it further. This substitution helps us relate the expression to the middle term $B$.

Step 5: Calculate $2B$

Next, we calculate two times the middle term, $2B$. This is the expression we need to show is equal to $A+C$.

Step 6: Substitute $x+z = 2y$ into $2B$

We substitute $x+z = 2y$ into the expression for $2B$. We can rewrite $2xz$ as part of $(x+z)^2$. This allows us to express $2B$ in terms of $x$, $z$, and $y$.

Step 7: Compare $A+C$ and $2B$

We have $A+C = x^2+xy+3y^2+yz+z^2$ and $2B = x^2+4y^2+z^2$. These expressions do not immediately look equal. Let's re-evaluate the substitution in $A+C$ more carefully.

Step 8: Re-evaluate $A+C$ using $x+z=2y$

Let's re-substitute $x+z=2y$ into the expression for $A+C$. We group the terms involving $y$ and $x+z$ to make the substitution clearer. This leads to $A+C = x^2+4y^2+z^2$.

Step 9: Conclusion

Since we have shown that $A+C = x^2+4y^2+z^2$ and $2B = x^2+4y^2+z^2$, it follows that $A+C = 2B$. This proves that the given terms are in an A.P.

More from this chapter

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• is 78 atom of the AP 5 11 17 23
• which time of Ap 3,8,13,18 is 78
• the sum of the first 15 terms of an ap is 450 and its first term is 10 find its common difference
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• the fifth time of an ap is 18 and 11th term is 42 find first term
• if the sum of the first n terms of an ap is S sem is equal to 3 and square + 5 and find its common difference

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