Tamilnadu Board Class 10 Maths Solutions Chapter 2 Numbers and Sequences Ex 2.9

Tamilnadu State Board Class 10 Maths Solutions Chapter 2 Numbers and Sequences Ex 2.9

Question 1.
Find the sum of the following series
(i) 1 + 2 + 3 + … + 60
(ii) 3 + 6 + 9 + … + 96
(iii) 51 + 52 + 53 + … + 92
(iv) 1 + 4 + 9 + 16 + … + 225
(v) 6² + 7² + 8² + … + 21²
(vi) 10³ + 11³ + 12³ + … + 20³
(vii) 1 + 3 + 5 + … + 71
Solution:
(i) 1 + 2 + 3 + … + 60

(ii) 3 + 6 + 9 + … + 96

(iii) 51 + 52 + 53 + … + 92

(iv) 1 + 4 + 9 + 16 + … + 225


(v) 6² + 7² + 8² + … + 21²

(vi) 10³ + 11³ + 12³ + … + 20³

(vii) 1 + 3 + 5 + … + 71 = n²

Question 2.
If 1 + 2 + 3 + … + k = 325, then find 1³ + 2³ + 3³ + … + K³.
Solution:
1 + 2 + 3 + … + K = 325

If 1 + 2 + 3 … + k = 325
1³ + 2³ + 3³ + … + K³ = (325)² = 105625

Question 3.
If 1³ + 2³ + 3³ + … + K³ = 44100 then find 1 + 2 + 3 + … + k.
Solution:
If 1³ + 2³ + 3³ + … + K³ = 44100
1 + 2 + 3 + … + K = \(\sqrt { 44100 }\)
= 210

Question 4.
How many terms of the series 1³ + 2³ + 3³ + … should be taken to get the sum 14400?
Solution:
1³ + 2³ + 3³ + … + n³ = 14400
\(\left(\frac{n(n+1)}{2}\right)^{2}\) = 14400 = (120)2
\(\frac{n(n+1)}{2}\) = \(\sqrt { 14400 }\) = 120
n(n + 1) = 240
Method 1:
n² + n – 240 = 0
n² + I6n – 15n – 240 = 0
n(n + 16) – 15(n + 16) = 0
(n + 16)(n – 15) = 0
n = -16, 15
∴ 15 terms to be taken to get the sum 14400.
Method 2:
n² + n – 240 = 0

Question 5.
The sum of the squares of the first n natural numbers is 285, while the sum of their cubes is 2025. Find the value of n.
Solution:
1² + 2² + 3² + …… + n² = 285
1³ + 2³ + 3³ + …… + n³ = 2025

Question 6.
Rekha has 15 square colour papers of sizes 10 cm, 11 cm, 12 cm,…, 24 cm. How much area can be decorated with these colour papers?
Solution:
10² + 11² + 12² + … + 242
= (1² + 2² + … + 24²) – (1² + 2² + … + 9²)

∴ Rekha has 4615 cm² colour papers. She can decorate 4615 cm² area with these colour papers.

Question 7.
Find the sum of the series (2³ – 1) + (4³ – 3³) + (6³ – 15³) +… to (i) n terms (ii) 8 terms.
Solution:
(2³ – 1) + (4³ – 3³) + (6³ – 15³) + … n

= 2048 + 192 = 2240

Samacheer Kalvi 10th Maths Guide