ALTERNATING  POINT SEQUENCES

We can also generate interesting results for for Fibonacci sequences (and extensions) when we alternately add and subtract each term.

Thus for the Fibonacci sequence

1/109 = .01 - .001 + .0002 - .00003 + .000005 - .0000008 + .00000013 - .000000021 + .0000000034 -…..

Whereas for the case where all signs are positive the total value of the sequence = the reciprocal 100 - 10 - 1, in this case where the signs alternate the total value = the reciprocal of 100 + 10 - 1.

In general terms for extended sequences where the value for the last term has the value of a (with b = 1) then the total value of sequence = the reciprocal of
100 + 10a - 1.

Thus for the Pell series (with a = 2, b = 1) the value of sequence of terms = the reciprocal of 100 + 20 - 1 = 119.

Thus 1/119 = .01 - .002 + .0005 - .00012 + .000029 - .0000070 + .00000169 - ….

Again this applies across bases n (where b <  n).

Thus in base 8 the Pell sequence with alternating terms sums to the reciprocal of 100 + 20 - 1 = 117

Thus (in base 8)

1/117 = .01 - .002 + .0005 - .00014 + .000035 - 0000106 + .00000211 -…

We can also derive similar type results when we take multiples of first term (b) while holding the second term constant.

Thus when we take twice the first term when adding to the second term we get

1, 1, 3, 5, 11, 21, 43, 85, 171, 341….

The total value of this sequence (with alternating signs) = reciprocal of 100 + 10 - 2 = 108.

Thus 1/108 = .01 - .001 + .0003 - .00005 + .000011 - .0000021 + .00000043 - .000000085 +…

So in general terms where multiples of the 1st term (b) are added to the second to derive the sequence, the total of the point sequence (with alternating signs) = the reciprocal of 100 + 10 - b. (where b < the number base n).

So in base 10, where b = 9, we obtain the sequence

0, 1, 1, 10, 19, 109, 270, 1251, ….

Thus the total of terms of decimal point sequence (with alternating signs) = the reciprocal of 101.

Thus 1/101 = .01 - .001 + .0010 - .00019 + .000109 - .0000270 + .00001251 - …

We can of course combine multiples of a and b

Thus in general terms the total of terms of point sequence (with alternating signs) = reciprocal of 100 + 10a - b.

Thus where a = 3 and b = 2 we obtain the sequence

This series is 0, 1, 3, 11, 39, 139, 495, 1763,…

Thus the total for the related point series (with alternating signs) = 100 + 30 - 2 = 128.

Thus 1/128 = .01 - .003 + .0011 - .00039 + .000139 - .0000495 + .00001763 -….

Again this result is valid across bases n (where a and b <  n).