Given the pattern you want to follow, the two formulas needed are pretty simple.
Start by setting up your table like the one below:

Your table should have one header column (A) and two header rows (1 and 2).
B1 is the only cell in column 1 to contain a formula:
B1: "Wk "&WEEKNUM(B2,1)
Note that the result is week number 47—true for either setting of the last argument, 1 (Sunday is 1) or 2 (Monday is 1)
B2 contains a Date and time value, with only the month and day number entered as shown.
Columns C and D will contain data, and should be left empty for the moment.
Column E contains three formulas:
E1: "Wk "&WEEKNUM(E2,1)
This can be entered as shown, or you can select cell B1 with one click, copy, then select E1 with one click and paste. Numbers will automatically change the column reference to match the new location.
E2: B2+7
This returns a date and time value 7 days after the date in B2, formatted for the same appearance as B2.
E3: B3+C3-D3
This gets the previous week's balance from column B, adds the income amount from column C and subtracts the expense amount from column D and returns the current balance.
Fill this formula down as many rows as you have categories.
Your table should now look very much like the image above.
I uses shorter labels in the income and expense columns to allow compacting the table for display here.
You can put the "Nov 17 thru Nov 23" label in C1 if you wish, but it will not update in the next step, and I suspect that editing it would take more time than entering it, so I left these cells empty. I notice also that I reversed the row locations for In and Out labels.
The data I've entered (5000 in cell B3) can be omitted at this point. I placed it there only to check that the formula in column E was working.
Select all cells in columns C, D and E as shown below by clicking on C1, holding the mouse button down and dragging to the last cell in column E. Note that the selection does NOT highlight the column reference tabs above the table with the deep blue that indicates a selection of the columns.

With the cells in those three columns selected, bring the mouse pointer close to the middle of the right boundary of the selection rectangle (between columns E and F. A Fill handle (small yellow filled circle) will appear in that double blue line, near its centre. Place the mouse pointer on the handle, then click and drag right to copy the selected cells into the rest of the columns to the right. The weekly dates (and Week numbers) and totals will appear in columns H, L and N, as dots while the mouse button is down, and as seen below when you reach the end of the table and release the mouse button.
.
Fill in the categories in column A, and the amounts in the in and out columns. Take care to not enter data in the columns showing balances, as doing so will replace the formula in that cell.
Regards,
Barry
PS: Income and expense amounts should all be entered as positive values.