Number-series-a-3-2a-b-6-a-9-a-b-5-a-15-3a-b-7-Find-a-

Question Number 206365 by hardmath last updated on 12/Apr/24

Number series :

a_{3} = 2 a + b - 6

a_{9} = a + b + 5

a_{15} = 3 a + b - 7

Find : a = ?

Commented by A5T last updated on 12/Apr/24

A r i t h m e t i c p r o g r e s s i o n ? I s a t h e f i r s t t e r m ?

Commented by hardmath last updated on 12/Apr/24

yes, arithmetic

Commented by A5T last updated on 12/Apr/24

I s a t h e f i r s t t e r m ?

Commented by hardmath last updated on 12/Apr/24

yes dear ser

Answered by A5T last updated on 12/Apr/24

a_{15} - a_{3} = a - 1 = 12 d

a_{3} - a_{9} = a - 11 = - 6 d

\Rightarrow 18 d = 10 \Rightarrow d = \frac{5}{9} \Rightarrow a = \frac{23}{3}

a_{3} = a + 2 d = \frac{79}{9} = \frac{46}{3} - 6 + b \Rightarrow b = \frac{- 5}{9}

\Rightarrow a_{3} = \frac{79}{9}, a_{9} = \frac{109}{9}; a_{15} = \frac{139}{9}

a_{n} = \frac{64 + 5 n}{9}

Answered by Rasheed.Sindhi last updated on 13/Apr/24

a_{n} is an AP (g i v e n)

∵ 3, 9, 15 a r e i n AP

∴ a_{3}, a_{9}, a_{15} a r e a l s o i n AP

∴ a_{9} = \frac{a_{3} + a_{15}}{2}

a + b + 5 = \frac{5 a + 2 b - 13}{2}

5 a + 2 b - 13 = 2 a + 2 b + 10

3 a = 23

a = \frac{23}{3}