Question

If the first term of an AP is 2 and the sum of the first five terms is equal to one-fourth of the sum of the next five terms, then (i) show that a20=-112 (ii) find the sum of the first 30 terms. ​

Answer 1

Step-by-step explanation:

Let a be the first term

then a=2

sum of first five terms=5/2(4+4d)

= 5/2×4(1+d)

=10+10d

sum of first 10 terms=10/2(4+9d)

=5(4+9d)

=20+45d

for calculation sum of next 5 terms would be

S10-S5

=10+35d

acc to question

10+10d=1/4(10+35d)

40+40d=10+35d

5d=- 30

d=-6

sum of 30 terms=30/2(4-(29×6))

=15(-170)

=-2550

2nd part of the question

a20=a+19d

=2-19(5)

=2- 114

=-112

Hence proved

Answer 2

$\huge\mathfrak{Answer:}$

Solution 1.

Let "a" be the first term

then a=2

sum of first five terms=5/2(4+4d)

= 5/2×4(1+d)

=10+10d

sum of first 10 terms=10/2(4+9d)

=5(4+9d)

=20+45d

=> calculation of sum next 5 terms would be

S10-S5

=10+35d

acc to question

10+10d=1/4(10+35d)

40+40d=10+35d

5d=- 30

d=-6

sum of 30 terms=30/2(4-(29×6))

=15(-170)

=-2550

Solution 2 =

a20=a+19d

=2-19(5)

=2- 114

=-112

So , the required answer is( -112).