Observe the following pattern1 = \(\frac{1}{2}\){1 x (1 + )}1 + 2 =

1.	Observe the following pattern1 = \(\frac{1}{2}\){1 x (1 + )}1 + 2 = \(\frac{1}{2}\){2 x (2 + )}1 + 2 + 3 + 4 = \(\frac{1}{2}\){3 x (3 x 1)}1 + 2 + 3 + 4 = \(\frac{1}{2}\){4 x (4 x 1)}And find the values of each of the following:(i) 1+2+3+4+5+……….+50(ii) 31+32+……..+50
Answer» R.H.S = \(\frac{1}{2}\)[No.of terms in L.H.S x (No. of terms + 1)] (Therefore, only when L.H.S starts with 1) Therefore. (i) 1 + 2 + 3 +…..50 = \(\frac{1}{2}\)[50 x (50 + 1)} = 25 × 51 = 1275 (ii) 31 + 32 +…..+50 = (1 + 2 + 3 + …. + 50) – (1 + 2 + ….. 30) = 1275 - [\(\frac{1}{2}\)(30 x 30 + 1)] = 1275 – 465 = 810

Observe the following pattern1 = \(\frac{1}{2}\){1 x (1 + )}1 + 2 = \(\frac{1}{2}\){2 x (2 + )}1 + 2 + 3 + 4 = \(\frac{1}{2}\){3 x (3 x 1)}1 + 2 + 3 + 4 = \(\frac{1}{2}\){4 x (4 x 1)}And find the values of each of the following:(i) 1+2+3+4+5+……….+50(ii) 31+32+……..+50

Answer»

R.H.S = \(\frac{1}{2}\)[No.of terms in L.H.S x (No. of terms + 1)] (Therefore, only when L.H.S starts with 1)

Therefore.

(i) 1 + 2 + 3 +…..50 = \(\frac{1}{2}\)[50 x (50 + 1)}

= 25 × 51

= 1275

(ii) 31 + 32 +…..+50 = (1 + 2 + 3 + …. + 50) – (1 + 2 + ….. 30)

= 1275 - [\(\frac{1}{2}\)(30 x 30 + 1)]

= 1275 – 465

= 810