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Arithmetic Progression

If An= n, n...

Question

If $A_{n} = (n, n + 1)$ then $10 (A_{10} A_{11})^{2} + 11 (A_{11} + A_{12})^{2} + . . . . . . + 20 (A_{20} A_{21})^{2}$ is equal to

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Solution

As ${(A_{n} A_{n + 1})}^{2} = {((n, (n + 1) ((n + 1), (n + 2))))}^{2} = 2$
then
$10 (A_{10} A_{11})^{2} + 11 (A_{11} + A_{12})^{2} + . . . . . . + 20 (A_{20} A_{21})^{2} = 2 (10 + 11 + 12 + . . . + 20) = 530$

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