Question

if a,b and c are in Ap show that a+3k , b+3k and c +3k are in ap​

Answer 1

${\huge{\sf {\green{\underline{Answer}}}}}$

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Given,

a, b, c are in AP

So,

( c - b ) = ( b - a ) ........ (1)

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Now,

( b + 3k ) - ( a + 3k )

= b + 3k - a - 3k

= b - a + 3k - 3k

= b - a

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Again,

( c + 3k ) - ( b + 3k )

= c + 3k - b - 3k

= c - b - 3k + 3k

= c - b

= b - a [ From equation 1 ]

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Therefore,

( b + 3k ) - ( a + 3k ) = ( c + 3k ) - ( b - 3k )

So,

( a + 3k ) , ( b + 3k ) and ( c + 3k ) are in AP.

--------- ( Proved )

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${\huge{\sf {\pink{\underline{More \:\: Information :}}}}}$

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In an AP the terms have a common difference.

To find the common difference of an AP the term is subtracted from its succeeding term.

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Formula for Last term of an AP $( {a}_{n} )$

${a}_{n} = a \: + \: (n - 1) \: d$

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where,

First term = a

No. of terms = n

Common Difference = d

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Formula for sum of 'n' terms of an AP $( {S}_{n} )$

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If the last term is not known :

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${S}_{n} = \frac{n}{2} (2a + (n - 1)d)$

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where,

First term = a

No. of terms = n

Common Difference = d

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If the last term is known :

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${S}_{n} = \frac{n}{2} (a + l)$

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where,

First term = a

No. of terms = n

Common Difference = d

Last term = $l$