Wrong.
Shear does not accumulate like displacement through the length of the spring. If it did, there would be no reaction at one of the supports- either the spring perch or the LCA.
Viewing each coil as a free body is useful, and spring body connects the equal and opposite (in a static case) forces at the spring perch and LCA via shear.
Consider the spring again 10" 10 coils, 100lb/inch. For a 100lb load, each coil sees 100lb, and deflects .1". The spring constant of each coil is the force over the deflection, or 100/.1 = 1000lb/in.
Cut a coil off, 100lb load, deflection is 9x.1" or .9". To achieve 1" deflection, each coil deflects 1/9" = .111", the reaction is 1000*.111=111 lb. The new spring constant is 111 lb/in.
Another way to look at it is each coil being in series w/ the next.
http://en.wikipedia.org/wiki/Hooke%27s_law#Multiple_springs
Springs in series add like resistors in parallel:
If added in series, 1/k = Sum(1/k1 + 1/k2 ....)
10 coils 1/k = 10 * 1/1000 =.01
k = 1/.01 = 100
9 coils 1/k = 9 * 1/1000 = .009
k = 1/.009 = 111