Which of the following are APs? If they form an AP, find the common difference d and write three more terms.
2, 4, 8, 16, ...
Not an AP. Differences: 4-2=2, 8-4=4, 16-8=8. Since differences are not equal, it does not form an AP.
2, 5/2, 3, 7/2, ...
It is an AP. Common difference d = 1/2. Three more terms: 4, 9/2, 5.
-1.2, -3.2, -5.2, -7.2, ...
It is an AP. Common difference d = -2. Three more terms: -9.2, -11.2, -13.2.
-10, -6, -2, 2, ...
It is an AP. Common difference d = 4. Three more terms: 6, 10, 14.
3, 3 + , 3 + 2, 3 + 3, ...
It is an AP. Common difference d = . Three more terms: 3+4, 3+5, 3+6.
0.2, 0.22, 0.222, 0.2222, ...
Not an AP. Differences: 0.22-0.2=0.02, 0.222-0.22=0.002, 0.2222-0.222=0.0002. Since differences are not equal, it does not form an AP.
0, -4, -8, -12, ...
It is an AP. Common difference d = -4. Three more terms: -16, -20, -24.
-1/2, -1/2, -1/2, -1/2, ...
It is an AP. Common difference d = 0. Three more terms: -1/2, -1/2, -1/2.
Explanation
To determine if a sequence forms an AP, we check if the difference between consecutive terms (a_{k+1} - a_k) is constant. The textbook demonstrates this method in Example 2, showing that if a_2 - a_1 = a_3 - a_2 = a_4 - a_3, then the sequence is an AP with common difference d. When the common difference is found, three more terms are obtained by adding d successively to the last given term.