Only Mathsdude will answer this Question ....... The Pic is attached too
The sum of n terms of the series 1.2 + 3.2 ^ 2 + 5.2 ^ 3 + 7.2 ^ 4 +....... is
(a) (n - 1) * 2 ^ (n + 2) - 2 ^ (n + 1) + 6
(b) (n + 1) * 2 ^ (n + 2) - 2 ^ (n + 1) + 6
(c) (n - 1) * 2 ^ (n + 2) - 2 ^ (n + 1) - 6
(d) None
The sum of n terms of the series 1.2 + 3.2 ^ 2 + 5.2 ^ 3 + 7.2 ^ 4 +....... is
(a) (n - 1) * 2 ^ (n + 2) - 2 ^ (n + 1) + 6
(b) (n + 1) * 2 ^ (n + 2) - 2 ^ (n + 1) + 6
(c) (n - 1) * 2 ^ (n + 2) - 2 ^ (n + 1) - 6
(d) None
Answers & Comments
Verified answer
Answer:
[tex]\qquad\qquad\qquad\boxed{ \bf{ \:(d) \: \: None \: }} \\ \\ [/tex]
Step-by-step explanation:
Given series is
[tex]\sf \: 1.2 + 3. {2}^{2} + 5. {2}^{3} + 7. {2}^{4} + ... \: n \: terms \\ \\ [/tex]
Let assume that
[tex]\sf \: S = 1.2 + 3. {2}^{2} + 5. {2}^{3} + 7. {2}^{4} + ... \: n \: terms \\ \\ [/tex]
can be rewritten as
[tex]\sf \: S = 1.2 + 3. {2}^{2} + 5. {2}^{3} + 7. {2}^{4} + ... \: + (2n - 1) {2}^{n} \\ \\ [/tex]
[tex]\boxed{ \sf{ \: \because \: {n}^{th} \: term \: of \: 1,3,5... \: \: is \: \: \: 1 + (n - 1)2 = 2n - 1 \: }} \\ \\ [/tex]
Now, its a sum of product of corresponding terms of two progression
[tex]\sf \: 1,3,5,...,2n - 1 \: which \: is \: an \: AP \: series \\ \\ [/tex]
and
[tex]\sf \: 2, {2}^{2} , {2}^{3} ,..., {2}^{n} \: which \: is \: an \: GP \: series \\ \\ [/tex]
So, we have
[tex]\sf \: S = 1.2 + 3. {2}^{2} + 5. {2}^{3} + 7. {2}^{4} + ... \: + (2n - 1) {2}^{n} \\ \\ [/tex]
Multiply both sides by 2 (i.e. by common ratio), we get
[tex]\sf \: 2S = 1. {2}^{2} + 3. {2}^{3} + 5. {2}^{4} + ... \: + (2n - 1) {2}^{n + 1} \\ \\ [/tex]
On Subtracting above two equations, we get
[tex]\sf \: S = - 2 - 2. {2}^{2} - 2. {2}^{3} - 2. {2}^{4} - ... - 2. {2}^{n} \: + (2n - 1) {2}^{n + 1} \\ \\ [/tex]
[tex]\sf \: S = - 2 - {2}^{3} - {2}^{4} - {2}^{5} - ... \: - {2}^{n + 1} + (2n - 1) {2}^{n + 1} \\ \\ [/tex]
[tex]\sf \: S = - 2 - [ {2}^{3} + {2}^{4} + {2}^{5} + ... \: + {2}^{n + 1}] + (2n - 1) {2}^{n + 1} \\ \\ [/tex]
[tex]\sf \: S = - 2 - {2}^{2} \underbrace{ [ {2}^{1} + {2}^{2} + {2}^{3} + ... \: + {2}^{n - 1}] } + (2n - 1) {2}^{n + 1} \\ \\ \sf \: GP \: series \: with \: a = 2 \: and \: r = 2 \: and \: number \: of \: terms \: n - 1 \\ \\ [/tex]
[tex]\sf \: S = - 2 - {2}^{2} \bigg(\dfrac{2( {2}^{n - 1} - 1) }{2 - 1} \bigg) + (2n - 1) {2}^{n + 1} \\ \\ [/tex]
[tex]\sf \: S = - 2 - {2}^{3}( {2}^{n - 1} - 1) + (2n - 1) {2}^{n + 1} \\ \\ [/tex]
[tex]\sf \: S = - 2 - {2}^{n - 1 + 3} + {2}^{3} + (2n - 1) {2}^{n + 1} \\ \\ [/tex]
[tex]\sf \: S = - 2 - {2}^{n + 2} + 8 + (2n - 1) {2}^{n + 1} \\ \\ [/tex]
[tex]\sf\implies \sf \: S = 6 - {2}^{n + 2} + (2n - 1) {2}^{n + 1} \\ \\ [/tex]
Hence,
[tex]\qquad\qquad\qquad\boxed{ \bf{ \:(d) \: \: None \: }} \\ \\ [/tex]