The only way to do these is by using technology, such as a TI-84 or similar calculator.
Instead of doing your problems for you, I'll do a couple that are
exactly like yours and you can do yours the same exact way.
I'm using a TI-84.
This one is like your first one
(a) First term is 13 and the sum of the first 10 terms is 4543825.
Press Y= and make it read like this:
\Y1=13(X^10-1)/(
X-1)
\Y2=4543825
Then press WINDOW and make it read
WINDOW
Xmin=-10
Xmax=10
Xscl=1
Ymin=-10
Ymax=5000000 (Notice that I changed Ymax just larger than 4543825
Yscl=1
Xres=1
Press GRAPH
Press 2ND TRACE
(choose 5 intersect)
Press ENTER 3 times
Read at bottom of screen X=4 Y=4543825
Answer: r=4
You can do yours the same way.
-------------------------------------
This one is like your second one:
(b) first term is 2 and the sum of the first 7 terms is 2.856518
Press Y= and make it read like this:
\Y1=2(X^7-1)/(
X-1)
\Y2=2.856518
You don't need to set the window, just reset the window
and graph by pressing ZOOM and selecting 6:ZStandard
Graph will show
Press 2ND TRACE
(choose 5 intersect)
Press ENTER 3 times
Read at bottom of screen X=.3 Y=2.856518
Answer: r=0.3
You can do yours the same way.
Edwin