​ NOTE: In an expression like this ⇒ ∑^n_i = 1 X , n represents the upper limit, 1 represents the lower limit , ​x is the variable expression

Question

NOTE: In an expression like this ⇒ \(\sum ^n _i = 1 X\) , n represents the upper limit, 1 represents the lower limit , x is the variable expression which we are finding out the sum of and i represents the index of summarization.

(i) \(\displaystyle \sum_{n = 1}^{10}\) (2 + 3ⁿ)

(ii) \(\displaystyle \sum_{k = 1}^{n}\) [2^k + 3^{(k - 1)}]

(iii) \(\displaystyle \sum_{k = 1}^{n}\) 5ⁿ

Answer 1

(i) We can write this as (2 + 3¹ )+(2+3² ) + (2 +3³ )+… to 10 terms 

= ( 2+2+2+… to 10 terms) + ( 3+3²+3³+… to 10 terms) 

= 2×10 + (3+3²+3³+… to 10 terms) 

= 20 + (3+3²+3³+… to 10 terms)

Sum of a G.P. series is represented by the formula, S_n = a\(\frac{r^n - 1}{r-1}\), when r≠1. 

‘S_n’ represents the sum of the G.P. series up to nth terms, 

‘a’ represents the first term, 

‘r’ represents the common ratio and 

‘n’ represents the number of terms.

Here, 

a = 3 

r = (ratio between the n term and n-1 term) 3

n = 10 terms

Thus, sum of the given expression is 

= 20 + (3+32+33+… to 10 terms) 

= 20 + 88572 

= 88592

(ii) The given expression can be written as, 

(2¹ + 3¹-1 ) + (2² + 3²-1 ) + …to n terms 

= (2 + 3⁰ ) + (2²+ 3¹ ) + …to n terms 

= (2 + 1) +(2² + 3 ) + …to n terms

= (2 + 2² + …to \(\frac{n}{2}\) terms) + (1 + 3 + … to \(\frac{n}{2}\) terms)

Sum of a G.P. series is represented by the formula, S_n = a\(\frac{r^n - 1}{r-1}\), when r≠1. 

‘S_n’ represents the sum of the G.P. series up to nth terms, 

‘a’ represents the first term, 

‘r’ represents the common ratio and 

‘n’ represents the number of terms. 

Here, 

a = 2, 1 

r = (ratio between the n term and n-1 term) 2, 3

\(\frac{n}{2}\) terms

(iii) We can rewrite the given expression as 

(5¹ + 5² + 5³+ …to 8 terms)

Sum of a G.P. series is represented by the formula, S_n = a\(\frac{r^n - 1}{r-1}\) , when r>1. 

‘S_n’ represents the sum of the G.P. series up to nth terms, 

‘a’ represents the first term, 

‘r’ represents the common ratio and 

‘n’ represents the number of terms. 

Here, 

a = 5 

r = (ratio between the n term and n-1 term) 5 

n = 8 terms