Tamil Nadu Board of Secondary EducationHSC Arts Class 11th

# Compute the sum of first n terms of 1 + (1 + 4) + (1 + 4 + 42) + (1 + 4 + 42 + 43) + ... - Mathematics

Sum

Compute the sum of first n terms of 1 + (1 + 4) + (1 + 4 + 42) + (1 + 4 + 42 + 43) + ...

#### Solution

T1 = 1

T2 = 1 + 4

T3 = 1 + 4 + 42

Tn = 1 + 4 + 42 + ... n terms which is a G.P.

∴ Tn = (1(4^"n" - 1))/(4 - 1)

= (4^"n" - 1)/3

So, Sn = sum"t"_"n"

= (sum4^"n" - 1)/(sum3)

= (sum4^"n" - sum1)/(sum3)

sum4^"n" = 4^1 + 4^2 + ....... "n terms"

= (4(4^"n" - 1))/(4 - 1)

= (4(4^"n" - 1))/3

So (sum4^"n" - sum1)/(sum3)

= ((4(4^"n" - 1)- "n")/3)/3

= (4(4^"n" - 1) - 3"n")/9

= 4/9(4^"n" - 1) - "n"/3

Concept: Finite Series
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#### APPEARS IN

Tamil Nadu Board Samacheer Kalvi Class 11th Mathematics Volume 1 and 2 Answers Guide
Chapter 5 Binomial Theorem, Sequences and Series
Exercise 5.3 | Q 4 | Page 220
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