Question

In the following questions, two statements (i) and (ii) are given. Choose the valid statement.

1. 7 + 10.5 + 14 + ... + 91 = 1125
2. The number of terms in the progression 8 + 12 + 16 + ... + 124 is 20.

Options

• Only (i)

• Only (ii)

• Both (i) and (ii)

• Neither (i) nor (ii)

Answer 1

Neither (i) nor (ii)

Explanation:

(i) a = 7

d = 10.5 − 7

= 3.5

Last term (l) = 91

l = a + (n − 1)d

91 = 7 + (n − 1)3.5

91 − 7 = (n − 1)3.5

84 = (n − 1)3.5

`84/3.5 = (n − 1)`

24 = n − 1

24 + 1 = n

n = 25

Using the formula `S_n = n/2(a + l)`

`S_25 = 25/2(7 + 91)`

`S_25 = 25/2(98)`

S₂₅ = 25 × 49

S₂₅ = 1225

The sum is 1225, but the statement claims it is 1125. Therefore, Statement (i) is false.

(ii) a = 8

d = 12 − 8

= 4

Last term (l) = 124

124 = 8 + (n − 1)4

124 − 8 = (n − 1)4

116 = (n − 1)4

`116/4 = n − 1`

29 = n − 1

29 + 1 = n

n = 30

The actual number of terms is 30, but the statement claims it is 20. Therefore, Statement (ii) is false.