The value of n for which 704+12(704)+14(704)+⋯⋯=1984−12(1984)+14(1984)−⋯ up to n terms is

L.H.S=704+12(704)+14(704)+⋯n terms=704[1+12+14+⋯⋯n terms]=704{1[1−(12)n1−12]}=704×2[1−(12)n]→(1)R.H.S=1984−12(1984)+14(1984)⋯=1984[1−12+14⋯n terms]=1984[1(1−(−12)n1+12)]=1984×23×[1−(−12)n]→(2)L.H.S=R.H.S⇒704×2×[1−(12)n]=1984×23(1−(−12)n)⇒704×31984=1−(−12)n1−(12)n⇒3331=1−(−12)n1−(12)n⇒33−33(12)n=31−31(−12)n⇒33(12)n−31(−12)n=33−31⇒33(12)n−31(−12)=2→(3)It n is even12n=−12n⇒(33−31)12n=2⇒12n=1=120⇒n=0 It is not possibleCase –iiIt n is odd−12n=−12n(3)⇒3312n+3112n=2⇒6412n=2⇒(33+31)12n=2⇒12n=264=132=125⇒n=5

The value of n for which 704+12(704)+14(704)+⋯⋯=1984−12(1984)+14(1984)−⋯ up to n terms is

L.H.S=704+12(704)+14(704)+⋯n terms=704[1+12+14+⋯⋯n terms]=704{1[1−(12)n1−12]}=704×2[1−(12)n]→(1)R.H.S=1984−12(1984)+14(1984)⋯=1984[1−12+14⋯n terms]=1984[1(1−(−12)n1+12)]=1984×23×[1−(−12)n]→(2)L.H.S=R.H.S⇒704×2×[1−(12)n]=1984×23(1−(−12)n)⇒704×31984=1−(−12)n1−(12)n⇒3331=1−(−12)n1−(12)n⇒33−33(12)n=31−31(−12)n⇒33(12)n−31(−12)n=33−31⇒33(12)n−31(−12)=2→(3)It n is even12n=−12n⇒(33−31)12n=2⇒12n=1=120⇒n=0 It is not possibleCase –iiIt n is odd−12n=−12n(3)⇒3312n+3112n=2⇒6412n=2⇒(33+31)12n=2⇒12n=264=132=125⇒n=5

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The value of $n$ for which $704 + \frac{1}{2} (704) + \frac{1}{4} (704) + \dots \dots = 1984 - \frac{1}{2} (1984) + \frac{1}{4} (1984) - \dots$ up to n terms is

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Detailed Solution

L.H.S

$= 704 + \frac{1}{2} (704) + \frac{1}{4} (704) + \dots n terms$

$=704 [1 + \frac{1}{2} + \frac{1}{4} + \dots \dots n terms]$

$= 704 {1 [\frac{1 - {(\frac{1}{2})}^{n}}{1 - \frac{1}{2}}]}$

$= 704 \times 2 [1 - {(\frac{1}{2})}^{n}] \to (1)$

R.H.S

$= 1984 - \frac{1}{2} (1984) + \frac{1}{4} (1984) \dots$

$= 1984 [1 - \frac{1}{2} + \frac{1}{4} \dots n terms]$

$= 1984 [1 (\frac{1 - {(\frac{- 1}{2})}^{n}}{1 + \frac{1}{2}})]$

$= 1984 \times \frac{2}{3} \times [1 - {(\frac{- 1}{2})}^{n}] \to (2)$

$L.H.S = R.H.S$

$\Rightarrow 704 \times 2 \times [1 - {(\frac{1}{2})}^{n}] = 1984 \times \frac{2}{3} (1 - {(\frac{- 1}{2})}^{n})$

$\Rightarrow \frac{704 \times 3}{1984} = \frac{1 - {(\frac{- 1}{2})}^{n}}{1 - {(\frac{1}{2})}^{n}}$

$\Rightarrow \frac{33}{31} = \frac{1 - {(\frac{- 1}{2})}^{n}}{1 - {(\frac{1}{2})}^{n}}$

$\begin{array}{l} \Rightarrow 33 - 33 {(\frac{1}{2})}^{n} = 31 - 31 {(\frac{- 1}{2})}^{n} \\ \Rightarrow 33 {(\frac{1}{2})}^{n} - 31 {(\frac{- 1}{2})}^{n} = 33 - 31 \\ \Rightarrow 33 {(\frac{1}{2})}^{n} - 31 (\frac{- 1}{2}) = 2 \to (3) \end{array}$

It n is even

$\begin{array}{l} {(\frac{1}{2})}^{n} = {(\frac{- 1}{2})}^{n} \\ \Rightarrow (33 - 31) {(\frac{1}{2})}^{n} = 2 \\ \Rightarrow {(\frac{1}{2})}^{n} = 1 = {(\frac{1}{2})}^{0} \\ \Rightarrow n = 0 \end{array}$