Answer to Question #199734 in Calculus for Moe

Question

which of the expressions below are equal to

$3•4•8 - 5•6•16 + 7•8•32 - ...$

when summed to the $k$ terms?

1)

$k$$+ 1$

$Σ$ $a(a+1)(-1)$ ^a+1 $2^a$

$a =$ $3$

2)

$k$

$Σ$ $(2a + 1) (2a + 2) 2$ ^{a + 2}

$a =$ $1$

3)

$k$$+ 1$

$Σ$ $4(-1)^a a(2a-1)2^a$

$a =$ $2$

4)

$k$

$Σ$ $(-1)^a a(a+1)(a+5)$

$a =$ $3$

 

5) none of the above

Answer 1

Given, the expression

3•4•8−5•6•16+7•8•32−... (upto k times).

1)

$Σ_{a=3}^{k+1} a(a+1)(-1)^{a+1}2^a\\ =3.4.8-4.5.16+5.6.32-...(\text{upto k +1 terms})$

2)

$Σ_{a=1}^{k} (2a+1)(2a+2)2^{a+2}\\ =3.4.8+5.6.16+7.8.32+...(\text{upto k terms})$

3)

$Σ_{a=2}^{k+1}4(-1)^a a(2a-1)2^{a}\\ =4.2.3.4-4.3.5.8+4.4.7.16+...(\text{upto k +1 terms})$

4)

$Σ_{a=3}^{k}(-1)^a a(a+1)(a+5)\\ =-3.4.8+4.5.9-5.6.10+...(\text{upto k terms})\\ \text{Since, none of the above series matched woth the given series.}\\\text{ Therefore, correct option is (5) none of these.}$