Given: 12∑k=1log8ak=2010
⇒log8a1+log8a2+log8a3+⋯+log8a12=2010⇒log8[a1a2a3⋯a12]=2010⇒a.ar.ar2.ar3⋯ar11=82010⇒a12r66=26030⇒a2r11=21005
Let a=2α, r=2β, where α,β are non negative integers
⇒2α+11β=1005
If α=0, then
β=[100511]=91
⇒β≤91
Also, 11β=1005−2α
Which is odd, so β is also odd, so
β=1,3,5,7,...,91
Therefore, the number of terms
1+(n−1)×2=91⇒n=46
Hence, there are 46 pairs of (a,r) possible.