How does a cash deposit change the M1 measure of the money supply? The question asks what is the immediate effect of the cash deposit on the M1 measure of money supply. The official answer is "There is no change in the M1 measure of the money supply. (Demand deposits increase by the same amount that cash holdings fall.)." However, I think the some of the money deposited into the bank will be lend out and deposit into other banks and so on. So the multiplier also applies here, so the money supply should increase by the multiplier times the cash deposited minus the cash deposited(minus because cash holding has decreased). Why is that not true?

Please suggests if anything is not clear.

My guess: maybe it is because the question asks "immediate" effect and hence does not include the process of money being lend out again etc.?
It's best to always check the exact definitions for M questions, because they can vary a little between countries.  I'll use the US Federal Reserve's here, viz. "M1 is defined as the sum of currency held by the public and transaction deposits at depository institutions (which are financial institutions that obtain their funds mainly through deposits from the public, such as commercial banks, savings and loan associations, savings banks, and credit unions)."

The reason M1 doesn't increase is essentially semantic. Before the deposit occurs, M1 is the sum of currency held outside the banking system ("held by the public") and deposits at banks etc. Notice that currency held by the banks is not included in the M1 definition.

When the deposit occurs, the accompanying double entry book keeping is [debit cash, credit customer deposit]. Now the cash is held by the bank, so not included in M1, but the customer deposit that was created by the cash deposit is - so there's no change. This is the best way to handle this btw., since once the cash is deposited at a bank, it's no longer playing an active role in the money supply.

As far as the potential impact on the reserve requirement, and multiplier effect. As you're quite rightly suggesting, the question is trying to avoid that by saying "immediate". However, if you're in the USA, reserves no longer control the money supply the way the text book tries to describe, so there would not be expected to be a long term impact either. (reserve requirements only cover a fraction of bank deposits, and capital requirements actually dominate in terms of regulatory impact.)