Prove Ap+Ap+2q=2Ap+q
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Ap+ Ap =2ap Hence ap +ap +q=2ap +q
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Ap+Ap+2q=a+(p-1)d+a+(p+2q-1)d
=a+pd-d+a+pd+2qd-d
=2a+2pd-2qd-2d
=2(a+pd-qd-d)
=2[a+(p+q-1)d]........(1)
2Ap+q= 2[a+(p+q-1)d]........(2)
from(1)and(2),we get
Ap+Ap+2q=2Ap+q
=a+pd-d+a+pd+2qd-d
=2a+2pd-2qd-2d
=2(a+pd-qd-d)
=2[a+(p+q-1)d]........(1)
2Ap+q= 2[a+(p+q-1)d]........(2)
from(1)and(2),we get
Ap+Ap+2q=2Ap+q
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