Question

find the value of (1) 16+17+18+........+75
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Answer 1

Answer:

16+17+18...+75

this series in AP

a= 16

d = 1

an = 75

an = a+(n-1)d

75 =16+(n-1)1

59= n-1

n = 60

Sn = 60/2 [ 16+75]

= 30× 91

= 2730

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Answer 2

SOLUTION

TO DETERMINE

The value of

$\sf{16 + 17 + 18 + ......... + 75}$

EVALUATION

Here the given expression is

$\sf{16 + 17 + 18 + ......... + 75}$

First term = a = 16

Second Term = 17

Third term = 18

∴ Second Term - First term = Third term - Second Term

So this is an Arithmetic progression with

First term = a = 16

Common Difference = d = 17 - 16 = 1

Let 75 is the n th term of the Arithmetic progression

$\therefore \sf{ \: \: \: a + (n - 1)d = 75}$

$\implies \: \: \sf{16 + (n - 1) \times 1= 75}$

$\implies \: \: \sf{15 + n= 75}$

$\implies \: \: \sf{ n= 60}$

So there are 60 terms in the arithmetic progression

Hence the required sum

$\displaystyle \sf{ = \frac{n}{2} \bigg[ \:2a + (n - 1)d \bigg] }$

$\displaystyle \sf{ = \frac{60}{2} \bigg[ \:(2 \times 16) + (60 - 1) \times 1 \bigg] }$

$\displaystyle \sf{ =30 \times \bigg[ \:32 + 59\bigg] }$

$\displaystyle \sf{ =30 \times 91 }$

$= 2730$

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