Question

the sum of first 20 terms of an arithmetic progression is 50 and the sum of next 20 terms is -50 find the sum of first 100 terms of progressions

Answer 1

Heya

Refer to the attachment :)

• The answer is -750

Answer 2

The answer is -750

GIVEN

The sum of first 20 terms of an arithmetic progression is 50 and the sum of next 20 terms is -50.

TO FIND

Find the sum of first 100 terms of progressions.

SOLUTION

We can simply ssolve the above problem as follows;

We know that,

Sum of n terms of an AP is given by the formula;

$Sₙ = \frac{n}{2}(2a +(n - 1)d )$

Where,

n = number of terms

a = First term

d = common difference

Now,

Sum of 20 terms of AP = 50

So,

$50 = \frac{20}{2}(2a +(20 - 1)d )$

50 = 20a + 190d

5 = 2a + 19d (Equation 1)

Given,

Sum of next 20 terms = -50

$- 50 = \frac{20}{2}(2a +(20 + 20d + 19d )$

-50 = 10(2a+40d+19d)

-5 = 2a + 59d (Equation 2)

Subtracting (Equation 1) from (Equation 2)

-5-5 = 2a + 59d- ( 2a + 19d)

40d = -10

d = 1/4

Substituting the value of d in equation 1.

2a + 19d = 5

2a+ 19× 1/4 = 5

2a = 5-19/4

2a = 39/4

a = 39/8

Sum of first 100 terms;

$S₁₀₀ = \frac{100}{2} (2 \times \frac{39}{8} + 99 \times \frac{1}{4} )$

$= 50( \frac{39}{4} - \frac{99}{4} )$

$= 50 \times \frac{ - 60}{4}$

= -750

Hence, The answer is -750

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