Observe the following pattern12 = \(\frac{1}{6}\)[1 x (1 + 1) x (2 x

1.	Observe the following pattern12 = \(\frac{1}{6}\)[1 x (1 + 1) x (2 x 1 + 1)12 + 22 = \(\frac{1}{2}\)[2 x (2 + 1) x (2 x 2 + 1)]12 + 22 + 32 = \(\frac{1}{6}\)[3 x (3 + 1) x (2 x 3 + 1)]12 + 22 + 32 + 42 = \(\frac{1}{6}\)[4 x (4 + 1) x (2 x 4 + 1)]And find the values of each of the following.11 + 22 + 32 + 42 + .........+ 10251 + 62 + 72 + 82 + 92 + 102 + 112 + 122
Answer» R.H.S = \(\frac{1}{6}\)[ (No.of terms in L.H.S) x (No. + 1) x (2 x No. + 1)] (i) 1² + 2² + 3² + 4² + …… + 10² = \(\frac{1}{6} \) [10 (10 + 1) × (2 × 10 + 1)] = \(\frac{1}{6}\)[2310] = 385 (ii) 5² + 6² +….. + 12² = 1² + 2² + ….. 12² – (1² + 2² + 3³ + 4²) \(\frac{1}{6} \)[12 × (12 + 1) × (2 × 12 + 1)] -\(\frac{1}{6}\) [4 ×(4 + 1) × (2 × 4 + 1)] = 650 – 30 = 620

Observe the following pattern12 = \(\frac{1}{6}\)[1 x (1 + 1) x (2 x 1 + 1)12 + 22 = \(\frac{1}{2}\)[2 x (2 + 1) x (2 x 2 + 1)]12 + 22 + 32 = \(\frac{1}{6}\)[3 x (3 + 1) x (2 x 3 + 1)]12 + 22 + 32 + 42 = \(\frac{1}{6}\)[4 x (4 + 1) x (2 x 4 + 1)]And find the values of each of the following.11 + 22 + 32 + 42 + .........+ 10251 + 62 + 72 + 82 + 92 + 102 + 112 + 122

Answer»

R.H.S = \(\frac{1}{6}\)[ (No.of terms in L.H.S) x (No. + 1) x (2 x No. + 1)]

(i) 1² + 2² + 3² + 4² + …… + 10² = \(\frac{1}{6} \) [10 (10 + 1) × (2 × 10 + 1)]

= \(\frac{1}{6}\)[2310]

= 385

(ii) 5² + 6² +….. + 12² = 1² + 2² + ….. 12² – (1² + 2² + 3³ + 4²)

\(\frac{1}{6} \)[12 × (12 + 1) × (2 × 12 + 1)] -\(\frac{1}{6}\) [4 ×(4 + 1) × (2 × 4 + 1)]

= 650 – 30

= 620