What area of the brain experiences rapid growth of synapses and subsequent elimination of those synapses that are not used often during adolescence?

The correct answer is: prefrontal cortex

Synaptogenesis or formation of synapses begins about the 23rd week of gestation. The peak of synapse production is achieved at different periods. The massive overproduction of synapses that occurs in early childhood is followed by a gradual reduction of them also known as "synaptic pruning".  “Synaptic pruning” happens during childhood, and during adolescence.

The brain continues to grow and develop, but the overall number of neurons and synapses are reduced by up to 50%. This is process that is experience driven: unnecessary neuronal structures are removed and replaced by more complex and efficient structures, that are more required to the demands of adulthood.