Question

Find out which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference.$10+10+2^5,10+2^6,10+2^7,.....$

Answer 1

Given sequence is,

$10+10+2^5,10+2^6,10+2^7...\dots$

first term of this A.P is $a_1=10+10+2^5=52$

second term of this A.P is $a_2=10+2^6=74$

third term of this A.P is $a_3=10+2^7=138$

the condition for an sequence to be an A.P is their must be a common difference $(i.e.,d=a_{n+1}-a_n)$

putting n=1 in above equation

$d=a_2-a_1=74-52=22$

putting n=2 in above equation

$d=a_3-a_2=138-74=64$

as we can see we get two values of d , but in an A.P their must be a single value of d which means for this sequence we can't define a common difference

hence this sequence not form an A.P