To find the sum of the series A, we can rewrite it as:

A = 1/2 + 1/3 + 1/4 + ... + 1/9 + 1/10
A = 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10

Now, we can find a common denominator for all the fractions in the series:

A = (5+10+15+...+90+45)/90
A = (5(1+2+3+...+18)+45)/90
A = (5(18*19/2)+45)/90
A = (855+45)/90
A = 900/90
A = 10

Therefore, the sum of the series A is 10 cm.

Since a < 17/6 and A > 34/10, we can see that a must be less than 17/6 and A must be greater than 34/10. Given that A = 10, it satisfies the condition A > 34/10, but we need to find a value for a that is less than 17/6.

Let's find the value of a:

a = 17/6 - 10
a = 17/6 - 60/6
a = (17-60)/6
a = -43/6

Therefore, the value of a is -43/6, which is less than 17/6.