•Determine S S=a^2 +ar(a+r)+ar^2 (a+2r)+...+ar^(n−1) {a+(n−1) r}.


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Question Number 5742 by Rasheed Soomro last updated on 26/May/16
$•Determine S S=a^2 +ar(a+r)+ar^2 (a+2r)+...+ar^(n−1) {a+(n−1) r}.$
$∙ D e t e r m i n e S$ $S = a^{2} + a r (a + r) + {a r}^{2} (a + 2 r) + . . . + {a r}^{n - 1} {a + (n - 1) r} .$
	Answered by Yozzii last updated on 26/May/16
	$S=a^2 +ar(a+r)+ar^2 (a+2r)+ar^3 (a+3r)...+ar^(n−1) {a+(n−1)r} rS=a^2 r+ar^2 (a+r)+ar^3 (a+2r)+ar^4 (a+4r)+...+ar^(n−1) (a+(n−2)r)+ar^n (a+(n−1)r) rS=0+(a^2 r)+0+(a^2 r^2 )+0+(ar^3 )+0+(a^2 r^3 )+(2ar^4 )+a^2 r^4 +4ar^5 +...+(a^2 r^(n−1) )+(n−2)ar^n +a^2 r^n +ar^(n+1) (n−1) S=a^2 +(a^2 r)+ar^2 +(a^2 r^2 )+(2ar^3 )+(a^2 r^3 )+(3ar^4 )+...+(a^2 r^(n−1) )+(n−1)ar^n ⇒(r−1)S=−a^2 −ar^2 −ar^3 −ar^4 −...−ar^n −ar^(n+1) +a^2 r^n +nar^(n+1) (r−1)S=a^2 r^n −a^2 −ar^2 (1+r+r^2 +...+r^(n−2) +r^(n−1) )+nar^(n+1) (r−1)S=a^2 (r^n −1)+nar^(n+1) −((ar^2 (r^n −1))/((r−1))) S=((r^n −1)/(r−1))(a^2 −((ar^2 )/(r−1)))+((nar^(n+1) )/(r−1)) S=((nar^(n+1) )/(r−1))+((a(r^n −1)(a(r−1)−r^2 ))/((r−1)^2 )) S=((nar^(n+1) +a^2 (r^n −1))/(r−1))−((ar^2 (r^n −1))/((r−1)^2 )) If r<1 S can be divergent e.g let r=−56.$
	$S = a^{2} + a r (a + r) + {a r}^{2} (a + 2 r) + {a r}^{3} (a + 3 r) . . . + {a r}^{n - 1} {a + (n - 1) r}$ $r S = a^{2} r + {a r}^{2} (a + r) + {a r}^{3} (a + 2 r) + {a r}^{4} (a + 4 r) + . . . + {a r}^{n - 1} (a + (n - 2) r) + {a r}^{n} (a + (n - 1) r)$ $r S = 0 + (a^{2} r) + 0 + (a^{2} r^{2}) + 0 + ({a r}^{3}) + 0 + (a^{2} r^{3}) + (2 {a r}^{4}) + a^{2} r^{4} + 4 {a r}^{5} + . . . + (a^{2} r^{n - 1}) + (n - 2) {a r}^{n} + a^{2} r^{n} + {a r}^{n + 1} (n - 1)$ $S = a^{2} + (a^{2} r) + {a r}^{2} + (a^{2} r^{2}) + (2 {a r}^{3}) + (a^{2} r^{3}) + (3 {a r}^{4}) + . . . + (a^{2} r^{n - 1}) + (n - 1) {a r}^{n}$ $\Rightarrow (r - 1) S = - a^{2} - {a r}^{2} - {a r}^{3} - {a r}^{4} - . . . - {a r}^{n} - {a r}^{n + 1} + a^{2} r^{n} + {n a r}^{n + 1}$ $(r - 1) S = a^{2} r^{n} - a^{2} - {a r}^{2} (1 + r + r^{2} + . . . + r^{n - 2} + r^{n - 1}) + {n a r}^{n + 1}$ $(r - 1) S = a^{2} (r^{n} - 1) + {n a r}^{n + 1} - \frac{{a r}^{2} (r^{n} - 1)}{(r - 1)}$ $S = \frac{r^{n} - 1}{r - 1} (a^{2} - \frac{{a r}^{2}}{r - 1}) + \frac{{n a r}^{n + 1}}{r - 1}$ $S = \frac{{n a r}^{n + 1}}{r - 1} + \frac{a (r^{n} - 1) (a (r - 1) - r^{2})}{{(r - 1)}^{2}}$ $S = \frac{{n a r}^{n + 1} + a^{2} (r^{n} - 1)}{r - 1} - \frac{{a r}^{2} (r^{n} - 1)}{{(r - 1)}^{2}}$ $I f r < 1 S c a n b e d i v e r g e n t e . g l e t r = - 56 .$
	Commented by Rasheed Soomro last updated on 26/May/16

	$T h a n k S!$
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