Blog

Examples 26 to 30 of Fundamental(Numbers)

No Comments

(26) Find the sum of all odd numbers up to 100.

Soln. The given numbers are 1, 3, 5, 7, …, 99.

∴ $T_n$ = 2n – 1 ⇒ 2n – 1 = 99

∴ n = 50

∴ Required sum = $\frac{n}{2} (first \ term + last\ term)$

= $\frac{50}{2} \times (1 + 99)$

= 2500.

We know that,

$\sum{(2n-1)}$

= 1 + 3 + 5 + . . . + (2n – 1)

= n^2

Here, n = 50 (means total terms = 50)

∴ Required sum = 50² = 2500.

(27) Find the sum of all 2 digit numbers divisible by 6.

Soln. All 2 digit numbers divisible by 3 are : 12, 18, …, 96.

∴ = 6n + 6 ⇒ 6n + 6 = 96

∴ n = 15

∴ Required sum = $\frac{n}{2} ( a + l )$

= $\frac{15}{2}$ × ( 12 + 96 )

= 810

(28) How many terms are there in 2, 4, 8, 16, …, 4096 ?

Soln. Clearly 2, 4, 8, 16, …, 1024 form a G.P. with a = 2 and r = $\frac{4}{2}$ = 2.

Let the number of terms be n. Then,

ar^{n – 1}= 2 × 2^n-1 = 4096

∴ 2^n-1 = 2048 = 2¹¹.

∴ n – 1 = 11

∴ n = 12.

(29) 2 + 2² + 2³ + … + 2¹⁰ = ?

Soln. Given series is a G.P. with a = 2, r = 2 and n = 10.

Sum = $\frac{a(r^n-1)}{(r-1)}$

= $\frac{2\times(2^{10}-1)}{(2-1)}$

= ( 2 × 1023 )

= 2046.

(30) Find the number of factors of 10800.

Soln. 10800 = 2⁴ × 3³ × 5²

∴ The number of factors of 10800 = ( 4 + 1 ) · ( 3 + 1 ) · ( 2 + 1 )

= 5 · 4 · 3

= 60