What do I do with these christoffel symbols?

Sparkyboy
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Hey guys I'm a bit new to GR and stuck on this question? :/. So we are given that:

d2xi/dλ2+\Gammaijk dxi/dλ dxj/dλ = 0

and asked to show that d/dλ(gijdxi/dλdxj/dλ) = 0

So I expanded using the product rule to get:

\Gammaijkd2xi/dλ2 dxj/dλ +\Gammaijk dxi/dλd2 xj/dλ2

Then rearranged the first equation to get:

d2xi/dλ2 = - \Gammaijkdxi/dλ dxj/dλ

and substituted for the second order differential equations. That's where I get stuck as I don't know how to get rid of the christoffel symbols. I read somewhere that they have a high degree of symmetry - so maybe I can change the dummy indices to get a symmetric form and they cancel? Very confused - any help would be appreciated.
 
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I believe you are missing a term d(gij)/dλ*...?

I guess I don't follow exactly your "expand" term. If you skipped a lot of steps, I can't do them in my head haha.

There's something wrong with the indices though, you have two of the same indices "upstairs" (implied sum or not?), which almost never happens.
 
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