Explanation:

Sum of first n natural numbers = $\dfrac{n × (n + 1)}{2}$
Sum of first 17 natural numbers = $\dfrac{17 × 18}{2}$
= 17 × 9
= 153

Explanation:

Sum of first n natural numbers = $\dfrac{n × (n + 1)}{2}$
Sum of first 25 natural numbers = $\dfrac{25 × (25 + 1)}{2}$
= $\dfrac{25 × 26}{2}$
= 25 × 13
= 325

Explanation:

Sum of first n natural numbers = $\dfrac{n × (n + 1)}{2}$
Sum of first 25 natural numbers = $\dfrac{100 × (100 + 1)}{2}$
= $\dfrac{100 × 101}{2}$
= 50 × 101
= 5050

Explanation:

Sum of first 25 even numbers = n × (n + 1) = 25 × 26 = 650

Explanation:

Sum of first 100 even numbers = n × (n + 1) = 100 × 101 = 10100

Explanation:

Sum of first 20 odd numbers = 20 × 20 = 400

Explanation:

Sum of first 500 odd numbers = 500 × 500 = 250000

Explanation:

Arithmetic progression (AP)
a = 7 and d= 6
a, (a + d), (a + 2d), (a + 3d),(a + 4d)........
7, 13, 19, 25, (7+4(6))
7, 13, 19, 25,31

Explanation:

a = 7
d = 13 – 7 = 6
n=16
16$^{th}$term, t$_{16}$ = a + (n-1)d = 7 + (16 – 1)6 = 7 + 90 = 97
16$^{th}$term = 97

Explanation:

a = 6
l = 30
d = 9 – 6 = 3
n = $n = \dfrac{(l - a)}{d} + 1$
= $\dfrac{(30 - 6)}{3} + 1 $
= $\dfrac{24}{3} + 1 $
= 8 + 1
= 9
Sum, S
=$\dfrac{n}{2}(a+l)$
=$\dfrac{9}{2}(6+30)$
=$\dfrac{9}{2} \times 36 $
=$9 \times 18$
=162

Explanation:

a = 5, r - $\dfrac{15}{5}$ = 3, n=5
5^th term, t₅
=$ar^{n-1} = 5 \times 3^{5-1}$
= $5 \times 3^4 = 5 \times 81$ = 405

Explanation:

a = 1, r =$\dfrac{\left(\dfrac{1}{2}\right)}{1} $= $\dfrac{1}{2}$, n = 5

Here r < 1. Hence,

$S_6$ = $ \dfrac{a(1 - r^n)}{1 - r} $= $ \dfrac{1\left[1 - \left(\dfrac{1}{2}\right)^5 \right]}{\left(1 - \dfrac{1}{2}\right)} $

= $\dfrac{\left(1 - \dfrac{1}{32} \right)}{\left(\dfrac{1}{2}\right)}$ = $\dfrac{\left(\dfrac{31}{32}\right) }{\left(\dfrac{1}{2}\right)} $=$ \dfrac{31}{16}$ = $1\dfrac{15}{16}$