# Arithmetic Progressions

#### Question 10:

Show that *a*_{1}, *a*_{2 }… , *a*_{n}
, … form an AP where *a*_{n} is defined as
below

(i) *a*_{n}
= 3 + 4*n*

(ii) *a*_{n}
= 9 − 5*n*

Also find the sum of the first 15 terms in each case.

#### Answer:

(i) *a*_{n}
= 3 + 4*n*

*a*_{1}
= 3 + 4(1) = 7

*a*_{2}
= 3 + 4(2) = 3 + 8 = 11

*a*_{3}
= 3 + 4(3) = 3 + 12 = 15

*a*_{4}
= 3 + 4(4) = 3 + 16 = 19

It can be observed that

*a*_{2}
− *a*_{1} = 11 − 7 = 4

*a*_{3}
− *a*_{2} = 15 − 11 = 4

*a*_{4}
− *a*_{3} = 19 − 15 = 4

i.e.,
*a*_{k}_{ + 1} − *a*_{k}
is same every time. Therefore, this is an AP with co...

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