Class 11 RD Sharma Solutions - Chapter 21 Some Special Series- Exercise 21.2

Sum the following series to n terms:

Question 1. 3 + 5 + 9 + 15 + 23 + . . . . n terms

Solution:

We are given the series: 3 + 5 + 9 + 15 + 23 + . . . . n terms.

Let’s take S as the sum of this series. Therefore,

S = 3 + 5 + 9 + 15 + 23 + . . . . + a_n–1 + a_n  . . . .(1)

S = 3 + 5 + 9 + 15 + 23 + . . . . + a_n–2 + a_n–1 + a_n  . . . .(2)

On subtracting (2) from (1), we get

=> S – S = [3 + (5 + 9 + 15 + 23 + . . . . + a_n–1 + a_n)] – [(3 + 5 + 9 + 15 + 23 + . . . . + a_n–2 + a_n–1) +a_n]

=> 0 = 3 + [(5 – 3) + (9 – 5) + (15 – 9) + . . . . + (a_n – a_n–1)] – a_n

=> a_n = 3 + [2 + 4 + 6 + . . . . (n–1) terms]

As the series 2+ 4 + 6 + . . . . (n–1) terms is an A.P., with first term(a) = 2 and common difference(d) = 4–2 = 2. So, we get,

=> a_n = 3 + (n–1) [2(2)+(n–2)2]/2      

= 3+ (n–1) [4+2n–4]/2

= 3 + n(n–1)

= n² – n+3

Now we have to summate our n^th term to find the sum(S_n) of this series.

= 

= 

= 

= 

= 

Therefore, sum of the given series up to n terms is .

Question 2. 2 + 5 + 10 + 17 + 26 + . . . . n terms

Solution:

We are given the series: 2 + 5 + 10 + 17 + 26 + . . . . n terms.

Let’s take S as the sum of this series. Therefore,

S = 2 + 5 + 10 + 17 + 26 + . . . . + a_n–1 + a_n  . . . .(1)

S = 2 + 5 + 10 + 17 + 26 + . . . . + a_n–2 + a_n–1 + a_n  . . . .(2)

On subtracting (2) from (1), we get

=> S – S = [2 + (5 + 10 + 17 + 26 + . . . . + a_n–1 + a_n)] – [(2 + 5 + 10 + 17 + 26 + . . . . + a_n–2 + a_n–1) +a_n]

=> 0 = 2 + [(5 – 2) + (10 – 5) + (17 – 10) + . . . . + (a_n – a_n–1)] – a_n

=> a_n = 2 + [3 + 5 + 7 + . . . . (n–1) terms]

As the series 3 + 5 + 7 + . . . . (n–1) terms is an A.P., with first term(a) = 3 and common difference(d) = 5–3 = 2. So, we get,

=> a_n = 2 + (n–1) [2(3)+(n–2)2]/2      

= 2+ (n–1) [6+2n–4]/2

= 2 + (n–1)(n+1)

= n² – 1 + 2

= n² + 1

Now we have to summate our n^th term to find the sum(S_n) of this series.

= 

= 

= 

Therefore, sum of the given series up to n terms is .

Question 3. 1 + 3 + 7 + 13 + 21 + 31 +  . . . . n terms

Solution:

We are given the series: 1 + 3 + 7 + 13 + 21 + 31 +  . . . . n terms.

Let’s take S as the sum of this series. Therefore,

S = 1 + 3 + 7 + 13 + 21 + 31 +  . . . . + a_n–1 + a_n  . . . .(1)

S = 1 + 3 + 7 + 13 + 21 + 31 +  . . . . + a_n–2 + a_n–1 + a_n  . . . .(2)

On subtracting (2) from (1), we get

=> S – S = [1 + (3 + 7 + 13 + 21 + 31 +  . . . . + a_n–1 + an)] – [(1 + 3 + 7 + 13 + 21 + 31 +  . . . . + a_n–2 + a_n–1) + a_n]

=> 0 = 1 + [(3 – 1) + (7 – 3) + (13 – 7) + . . . . + (a_n – a_n–1)] – a_n

=> a_n = 1 + [2 + 4 + 6 + . . . . (n–1) terms]

As the series 2 + 4 + 6 + . . . . (n–1) terms is an A.P., with first term(a) = 2 and common difference(d) = 4–2 = 2. So, we get,

=> a_n = 1 + (n–1) [2(2)+(n–2)2]/2      

= 1+ (n–1) [4+2n–4]/2

= 1 + n(n–1)

= n² – n+1

Now we have to summate our n^th term to find the sum(S_n) of this series.

= 

= 

= 

= 

= 

Therefore, sum of the given series up to n terms is .

Question 4. 3 + 7 + 14 + 24 + 37 +  . . . . n terms 

Solution: 

We are given the series: 3 + 7 + 14 + 24 + 37 +  . . . . n terms.

Let’s take S as the sum of this series. Therefore,

S = 3 + 7 + 14 + 24 + 37 +  . . . . + a_n–1 + a_n  . . . .(1)

S = 3 + 7 + 14 + 24 + 37 +  . . . . + a_n–2 + a_n–1 + a_n  . . . .(2)

On subtracting (2) from (1), we get

=> S – S = [3 + (7 + 14 + 24 + 37 +  . . . . + a_n–1 + a_n)] – [(3 + 7 + 14 + 24 + 37 +  . . . . + a_n–2 + a_n–1) + a_n]

=> 0 = 3 + [(7 – 3) + (14 – 7) + (24 – 14) + . . . . + (a_n – a_n–1)] – a_n

=> a_n = 3 + [4 + 7 + 10 + . . . . (n–1) terms]

As the series 4 + 7 + 10 + . . . . (n–1) terms is an A.P., with first term(a) = 4 and common difference(d) = 7–4 = 3. So, we get,

=> a_n = 3 + (n–1) [2(4)+(n–2)3]/2      

= 3+ (n–1) [8+3n–6]/2

= 3 + (n–1)(3n+2)/2

= 

= 

= 

Now we have to summate our n^th term to find the sum(S_n) of this series.

= 

= 

= 

= 

= 

Therefore, sum of the given series up to n terms is .

Question 5. 1 + 3 + 6 + 10 + 15 +  . . . . n terms

Solution:

We are given the series: 1 + 3 + 6 + 10 + 15 +  . . . . n terms.

Let’s take S as the sum of this series. Therefore,

S = 1 + 3 + 6 + 10 + 15 +  . . . . + a_n–1 + a_n  . . . .(1)

S = 1 + 3 + 6 + 10 + 15 +  . . . . + a_n–2 + a_n–1 + a_n  . . . .(2)

On subtracting (2) from (1), we get

=> S – S = [1 + (3 + 6 + 10 + 15 +  . . . . + a_n–1 + a_n)] – [(1 + 3 + 6 + 10 + 15 +  . . . . + a_n–2 + a_n–1) + a_n]

=> 0 = 1 + [(3 – 1) + (6 – 3) + (10 – 6) + . . . . + (a_n – a_n–1)] – a_n

=> a_n = 1 + [2 + 3 + 4 + . . . . (n–1) terms]

As the series 2 + 3 + 4 + . . . . (n–1) terms is an A.P., with first term(a) = 2 and common difference(d) = 3–2 = 1. So, we get,

=> a_n = 1 + (n–1) [2(2)+(n–2)1]/2      

= 1+ (n–1) [4+n–2]/2

= 1 + (n–1)(n+2)/2

= 

= 

= 

Now we have to summate our n^th term to find the sum(S_n) of this series.

= 

= 

= 

= 

= 

Therefore, sum of the given series up to n terms is .

Question 6. 1 + 4 + 13 + 40 + 121 +  . . . . n terms 

Solution:

We are given the series: 1 + 4 + 13 + 40 + 121 +  . . . . n terms.

Let’s take S as the sum of this series. Therefore,

S = 1 + 4 + 13 + 40 + 121 +  . . . . + a_n–1 + a_n  . . . .(1)

S = 1 + 4 + 13 + 40 + 121 +  . . . . + a_n–2 + a_n–1 + a_n  . . . .(2)

On subtracting (2) from (1), we get

=> S – S = [1 + (4 + 13 + 40 + 121 +  . . . . + a_n–1 + a_n)] – [(1 + 4 + 13 + 40 + 121 +  . . . . + a_n–2 + a_n–1) + a_n]

=> 0 = 1 + [(4 – 1) + (13 – 4) + (40 – 13) + . . . . + (a_n – a_n–1)] – a_n

=> a_n = 1 + [3 + 9 + 27 + . . . . (n–1) terms]

As the series 3 + 9 + 27 + . . . . (n–1) terms is a G.P., with first term(a) = 3 and common ratio(r) = 9/3 = 3. So, we get,

=> a_n = 1 + 3(3^n-1–1)/(3–1) 

= 1+ 3(3^n-1–1)/2 

= 1 + 3ⁿ/2 – 3/2

= 3ⁿ/2 – 1/2

Now we have to summate our n^th term to find the sum(S_n) of this series.

= 

= 

= 

= 

Therefore, sum of the given series up to n terms is .

Question 7. 4 + 6 + 9 + 13 + 18 +  . . . . n terms 

Solution:

We are given the series: 4 + 6 + 9 + 13 + 18 +  . . . . n terms.

Let’s take S as the sum of this series. Therefore,

S = 4 + 6 + 9 + 13 + 18 +  . . . . + a_n–1 + a_n  . . . .(1)

S = 4 + 6 + 9 + 13 + 18 +  . . . . + a_n–2 + a_n–1 + a_n  . . . .(2)

On subtracting (2) from (1), we get

=> S – S = [4 + (6 + 9 + 13 + 18 +  . . . . + a_n–1 + a_n)] – [(4 + 6 + 9 + 13 + 18 +  . . . . + a_n–2 + a_n–1) + a_n]

=> 0 = 4 + [(6 – 4) + (9 – 6) + (13 – 9) + . . . . + (a_n – a_n–1)] – a_n

=> a_n = 4 + [2 + 3 + 4 + . . . . (n–1) terms]

As the series 2 + 3 + 4 + . . . . (n–1) terms is an A.P., with first term(a) = 2 and common difference(d) = 3–2 = 1. So, we get,

=> a_n = 4 + (n–1) [2(2)+(n–2)1]/2

= 4 + (n–1)(n+2)/2

= 

= 

Now we have to summate our n^th term to find the sum(S_n) of this series.

= 

= 

= 

= 

= 

Therefore, sum of the given series up to n terms is .

Question 8. 2 + 4 + 7 + 11 + 16 +  . . . . n terms  

Solution:

We are given the series: 2 + 4 + 7 + 11 + 16 +  . . . . n terms.

Let’s take S as the sum of this series. Therefore,

S = 2 + 4 + 7 + 11 + 16 +  . . . . + a_n–1 + a_n  . . . .(1)

S = 2 + 4 + 7 + 11 + 16 +  . . . . + a_n–2 + a_n–1 + a_n  . . . .(2)

On subtracting (2) from (1), we get

=> S – S = [2 + (4 + 7 + 11 + 16 +  . . . . + a_n–1 + a_n)] – [(2 + 4 + 7 + 11 + 16 +  . . . . + a_n–2 + a_n–1) + a_n]

=> 0 = 2 + [(4 – 2) + (7 – 4) + (11 – 7) + . . . . + (a_n – a_n–1)] – a_n

=> a_n = 2 + [2 + 3 + 4 + . . . . (n–1) terms]

As the series 2 + 3 + 4 + . . . . (n–1) terms is an A.P., with first term(a) = 2 and common difference(d) = 3–2 = 1. So, we get,

=> a_n = 2 + (n–1) [2(2)+(n–2)1]/2

= 2 + (n–1)(n+2)/2

= 

= 

Now we have to summate our n^th term to find the sum(S_n) of this series.

= 

= 

= 

= 

= 

Therefore, sum of the given series up to n terms is .

Question 9.  

Solution: 

The n^th term of the given series would be,

a_n = 

= 

Now we have to summate our n^th term to find the sum(S_n) of this series.

= 

= 

= 

= 

Therefore, sum of the given series up to n terms is .

Question 10. to n terms.

Solution: 

We are given the n^th term of the given series,

a_n = 

= 

Now we have to summate our n^th term to find the sum(S_n) of this series.

= 

= 

= 

= 

Therefore, sum of the given series up to n terms is .