Sum to n term the following series : 1 - 2 3 + 3 9 - 4 27 + …

The given series 1 - 2 3 + 3 9 - 4 27 + … is arithmetic - geometric series each of whose term is obtained by multiplying the corresponding terms of A . P . 1 , 2 , 3 , 4 , … and G . P . 1 , - 1 3 , 1 9 , - 1 27 , … Let S n = 1 - 2 3 + 3 9 - 4 27 + … … + n terms We know that, the n t h term of a A . G . P . series is T n = [ a + ( n - 1 ) d ] r n - 1 . Here, n t h term of arithmetic geometric series be = [ 1 + ( n - 1 ) 1 ] - 1 3 n - 1 = n - 1 3 n - 1 ⇒ S n = 1 - 2 3 + 3 9 - 4 27 + … … + ( n - 1 ) - 1 3 n - 2 + n - 1 3 n - 1 (i) Multiply both sides with the common ratio of G . P . , r = - 1 3 ∴ - 1 3 S n = - 1 3 + 2 9 - 3 27 + … . . + ( n - 1 ) - 1 3 n - 1 + n - 1 3 n (ii) Now, equation (i) - equation (ii) gives 4 3 S n = 1 - 1 3 + 1 9 - 1 27 … + - 1 3 n - 1 - n - 1 3 n We know that, the sum of finite terms of a G . P . series is S n = a 1 - r n 1 - r , r < 1 . Here, Series on R.H.S. be an finite series (leaving last term) with a = 1 , r = - 1 3 = 1 1 - - 1 3 n 1 - - 1 3 - n - 1 3 n ⇒ 4 3 S n = 1 1 - - 1 3 n 1 - - 1 3 - n - 1 3 n ⇒ 4 3 S n = 3 4 1 - - 1 3 n - n - 1 3 n ⇒ S n = 9 16 1 - - 1 3 n - 3 n 4 - 1 3 n ∴ Sum to finite of the given series S n = 9 16 1 - - 1 3 n - 3 n 4 - 1 3 n .

Sum to n term the following series : 1 × 2 + 2 × 2 2 + 3 × 2 3 + 4 × 2 4 + …

The given series 1 × 2 + 2 × 2 2 + 3 × 2 3 + 4 × 2 4 + … is arithmetic - geometric series each of whose term is obtained by multiplying the corresponding terms of A . P . 1 , 2 , 3 , 4 , … and G . P . 2 , 2 2 , 2 3 , 2 4 , … Let S n = 1 × 2 + 2 × 2 2 + 3 × 2 3 + … . . + n terms We know that, the n t h term of a A . G . P . series is T n = [ a + ( n - 1 ) d ] r n - 1 . Here, n t h term of arithmetic geometric series be = [ 1 + ( n - 1 ) 1 ] 2 n - 1 = n · 2 n ⇒ S n = 1 × 2 + 2 × 2 2 + 3 × 2 3 + … . . + ( n - 1 ) 2 n - 1 + n 2 n (i) Multiply both sides with the common ratio of G . P . , r = 2 ∴ 2 S n = 1 × 2 2 + 2 × 2 3 + … + … + ( n - 1 ) 2 n + n 2 n + 1 (ii) Now, equation (i) - equation (ii) gives - S n = 1 × 2 + 1 × 2 2 + 1 × 2 3 + 10 4 … … + 1 × 2 n - n 2 n + 1 We know that, the sum of finite terms of a G . P . series is S n = a r n - 1 r - 1 , r > 1 . Here, Series on R . H . S . be an finite series (leaving last term) with a = 2 , r = 2 = 2 2 n - 1 2 - 1 - n 2 ⇒ - S n = 2 2 n - 1 2 - 1 - n 2 ⇒ - S n = ( 1 - n ) 2 n + 1 - 2 ⇒ S n = 2 + ( n - 1 ) 2 n + 1 ∴ Sum to finite of the given series S n = 2 + ( n - 1 ) 2 n + 1 .

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The sum to infinite terms of the series $1 + \frac{3}{4} + \frac{6}{4^{2}} + \frac{10}{4^{3}} + \frac{15}{4^{4}} + \dots$ is

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Important Questions on Sequence and Series

MEDIUM

Mathematics>Algebra>Sequence and Series>Arithmetico-Geometric Progression

If ${(20)}^{19} + 2 (21) {(20)}^{18} + 3 {(21)}^{2} {(20)}^{17} + . . . + 20 {(21)}^{19} = k {(20)}^{19}$ , then $k$ is equal to _____.

HARD

Mathematics>Algebra>Sequence and Series>Arithmetico-Geometric Progression

Let

S = 2 + \frac{6}{7} + \frac{12}{7^{2}} + \frac{20}{7^{3}} + \frac{30}{7^{4}} + \dots . .

then

4 S

is equal to

MEDIUM

Mathematics>Algebra>Sequence and Series>Arithmetico-Geometric Progression

Let

a_{1}, a_{2}, a_{3}, \dots

be an A.P. If

\sum_{r = 1}^{\infty} \frac{a_{r}}{2^{r}} = 4

, then

4 a_{2}

is equal to ______.

MEDIUM

Mathematics>Algebra>Sequence and Series>Arithmetico-Geometric Progression

For $k \in N$ , if the sum of the series $1 + \frac{4}{k} + \frac{8}{k^{2}} + \frac{13}{k^{3}} + \frac{19}{k^{4}} + . . . . . .$ is $10$ , then the value of $k$ is

MEDIUM

Mathematics>Algebra>Sequence and Series>Arithmetico-Geometric Progression

If the value of ${(1 + \frac{2}{3} + \frac{6}{3^{2}} + \frac{10}{3^{3}} + \dots . . upto \infty)}^{\log_{(0.25)} (\frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \dots . upto \infty)}$ is $l$ , then $l^{2}$ is equal to .