Will this lense focus on the Nikon D5100?

Www.ebay.co.uk%2Fitm%2FSigma-Ex-Apo-70-200Mm-1-2-8D-F-2-8-Lens-Nikon-Mount-%2F281387438259%3FssPageName%3DADME%253AB%253AWNA%253AGB%253A3160%26nma%3Dtrue%26si%3D4K1eAFuNG10G5z2mZ0i%25252F9Avp3Bg%25253D%26orig_cvip%3Dtrue%26rt%3Dnc%26_trksid%3Dp2047675.l2557&h=iAQGwCPsk

(Sigma 70 - 200 2.8 Hsm Nikon Mount)

If it's the Nikon mount it should. However I'd verify that with Sigma before you buy.

You have an entry level dSLR and it does not have an internal auto-focus motor which old Nikkor auto-focus lenses need to auto-focus.

This is why it is important that you buy the newer, faster auto-focusing Nikkor AF-S lenses.

If you decide to buy a third party lens, you need to assure yourself that it has an on-board auto-focus motor

I suggest that you visit the Sigma website and see if that specific lens has the internal motor you need so you can auto-focus with it on your camera.

If the lens is the Sigma 70-200 mm f/2.8 II APO EX DG HSM Macro offered on eBay UK for £475.00, if you read the description of the lens, here is what it says in part, "Equipped with a hyper sonic motor…" That specific lens will work as an auto-focus lens on your camera

If nothing else, ask the seller on eBay.