Question

1^2+2^2+3^2+.....+10^2= 385 find 2^2+4^2+6^2+....+20^2=

Answer 1

You are given the equation, $$1^2 + 2^2 + 3^2 +\cdots + 10^2 = 385.$$ From this equation, you want to find the value of $x$ in the following equation: $$2^2 + 4^2 + 6^2 +\cdots + 20^2 = x.$$ I know the problem you have been given does not include a variable $x$, but we will assign the unknown expression to such a variable in order to work it out algebraically. Now let's line up both the equations: $$\begin{align} 1^2 + 2^2 + 3^2 +\cdots + 10^2 &= 385\tag1 \\ 2^2 + 4^2 + 3^2 +\cdots + 20^2 &= x.\tag2\end{align}$$ Notice that every term being squared in equation $(2)$ is twice as large as every term being squared in equation $(1)$. To make that properly clear, let's re-write equation $(2)$ as follows: $$\begin{align} 1^2 + 2^2 + 3^2 +\cdots + 10^2 &= 385\tag1 \\ (2\times 1)^2 + (2\times 2)^2 + (2\times 3)^2 +\cdots + (2\times 10)^2 &= x.\tag2\end{align}$$ Now if we divide every squared term in equation $(2)$ by $2^2$, this will definitely equal equation $(1)$, which we observe being equal to $385$. We divide by $2^2$ because each term being squared is twice as large as each term being squared in equation $(1)$, and since we are squaring them, we must divide by $2^2$. But since we are adding up all the squared terms together, then by dividing each term by $2$, we are dividing the entire sum by $2$ as well. Now we let this entire sum be equal to $x$, so now we can write that $$\begin{align} \frac{x}{2^2} &= 385 \\ \\ \frac{x}{4} &= 385.\tag{since $2^2 = 4$}\end{align}$$ Remember that whatever you do to the left hand side of an equation, you must always do the same to the right hand side as well, in order for both sides to be equal. It is a bit like balanacing a seesaw, such that if you add weight on one side, you need to add the same amount of weight on the other side for the seesaw to be levelled.

Answer 2

Here ur answer mate..


2^2+4^2+6^2+....+20^2= "1540"

Hope it helps

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Have a good day..!!