Question

Find out which of the following sequences are arithmetic progressions. For those which ar3e arithmetic progressions, find out the common difference.
(i) $3,6,12,24\dots$
(ii) $0,-4,-8,-12\dots$
(iii) $\frac{1}{2},\frac{1}{4},\frac{1}{6}.\frac{1}{8},\dots$
(iv) $12,2,-8,-18,\dots$
(v) $3,3,33,3,\dots$
(vi) $p,p+90,p+180,p+270,\dots$ where $p=(999{)}^{999}$
(vii) $1.0,1.7,2.4,3.1,\dots$
(viii) $-225,-425,-625,-825,\dots$
(ix) $10+10+2^5,10+2^6,10+2^7,\dots$
(x) $a+b,(a+1)+b,(a+1)+(b+1),(a+2)+(b+1),(a+2)+(b+2),\dots$
(xi) $1^2,3^2,5^2,7^2,\dots$
(xii) $1^2,5^2,7^2,7^3,\dots$

Answer 1

Arithmetic Progression is the sequence of numbers such that the difference between the two successive terms is always constant. And that difference is called the Common Difference. It is also known as Arithmetic Sequence.

so according to the concept-

(i) 3,6,12,24 not an AP
(ii) 0,−4,−8,−12…IS AN AP Comman difference =-4-0=-4
(iii) 1/2,1/4,1/6.1/8,…not an AP
(iv) 12,2,−8,−18,…IS AN AP Comman difference =2-12=-10
(v) 3,3,33,3,…NOT AN AP
(vi) p,p+90,p+180,p+270,… where p=(999)^999 is an AP with comman difference=90+p-p=90
(vii) 1.0,1.7,2.4,3.1,…is an AP with comman difference 1.7-1=0.7

(viii) −225,−425,−625,−825, .....is an AP with comman difference -425-(-225)=-200
(ix) $10+10+2^5,10+2^6,10+2^7,\dots$not an AP

(x) a+b,(a+1)+b,(a+1)+(b+1),(a+2)+(b+1),(a+2)+(b+2),…ia an AP with comman difference
cd=(a+1)+b-a+b=1
(xi) $1^2,3^2,5^2,7^2,\dots$not an AP
(xii) $1^2,5^2,7^2,7^3,\dots$not an AP
simply check the comman difference.