A-series-of-natural-numbers-are-grouped-as-1-2-3-4-5-6-such-that-the-rth-group-contains-r-terms-Show-that-the-sum-of-the-numbers-in-the-2r-1-th-group-is-r

Question Number 148677 by nadovic last updated on 30/Jul/21

A series of natural numbers are

grouped as 1 + (2 + 3) + (4 + 5 + 6) + \dots

such that the r t h group contains r

terms . Show that the sum of the

numbers in the (2 r - 1) t h group is

r^{4} - {(r - 1)}^{4} .

Answered by gsk2684 last updated on 30/Jul/21

t_{1} = 1

t_{2} = ((1) + 1) + ((1) + 2)

t_{3} = ((1 + 2) + 1) + ((1 + 2) + 2) + ((1 + 2) + 3)

\dots .

t_{2 r - 1} =

((1 + 2 + 3 + . . + (2 r - 2) + 1) +

((1 + 2 + 3 + . . + (2 r - 2) + 2) +

((1 + 2 + 3 + . . + (2 r - 2) + 3) + \dots

((1 + 2 + 3 + . . + (2 r - 2) + 2 r - 1)

= (2 r - 1) (1 + 2 + 3 + . . + (2 r - 2)) + (1 + 2 + 3 + . . + (2 r - 1))

= (2 r - 1) \frac{(2 r - 2) (2 r - 1)}{2} + \frac{(2 r - 1) (2 r)}{2}

= (r - 1) {(2 r - 1)}^{2} + (2 r - 1) r

= (2 r - 1) {2 r^{2} - 3 r + 1 + r}

= (2 r - 1) (2 r^{2} - 2 r + 1)

r^{4} - {(r - 1)}^{4}

= (r^{2} - {(r - 1)}^{2}) (r^{2} + {(r - 1)}^{2})

= (2 r - 1) (2 r^{2} - 2 r + 1)

Commented by nadovic last updated on 30/Jul/21

T h a n k S i r

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