How can a new deposit of $10,000 at one bank create other new deposits at other banks? Suppose the desired reserve ratio is 10 percent and people keep no currency outside of the banks

What will be the new amount of deposits in the second and third rounds?

When the bank receives a new deposit of $10,000, it will loan some of the deposit. The proceeds of the loan will be deposited in another bank, which will then loan some of its new deposits. Hence an initial deposit at one bank leads to deposits at other banks.
With the desired reserve ratio of 10 percent, the first bank keeps $1,000 as reserves (10 percent of $10,000 ) and it can loan the remainder, $9,000. Thus the deposit at the second round bank will be $9,000. The second round bank keeps $900 as reserves (10 percent of $9,000 ) and can loan the remainder, $8,100. The deposit in the third round bank will be $8,100.