Find the sum of:
1 1 1 1
----- + ----- + ----- + ... + ----------
√1+√2 √2+√3 √3+√4 √999+√1000
The sum is one less than the square root of 1000.
This table illustrates the relationship
9 2.16227766016837952278706325 10.0000000000000000000000
99 9.00000000000000000000000000 100.0000000000000000000000
999 30.62277660168379256333537342 1000.0000000000000000000000
9999 99.00000000000000000000000000 10000.0000000000000000000000
99999 315.22776601683796116049052216 100000.0000000000000000000000
The last odd integer under the radical is in column 1 (column 3 is the last even integer, under the last radical, though programmed as the square of the sum incremented by 1). The middle column is the sum, which is 1 lower than the square root of the last column.
from
clc
for l=1:5
t=0;
for i0=10^l-1:-1:1
i=vpa(i0);
t=t+(1/(sqrt(i)+sqrt(i+1)));
end
fprintf('%8d %30.26f %30.22f\n',10^l-1, t, (t+1)^2)
end
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Posted by Charlie
on 2022-07-18 08:33:49 |
computer solution
