Arithmetic Progression

Here, the first term is \(a=70\) and the common difference is \(d=62-70=-8\). The \(n\)th term is \(a_n=a+(n-1)d\). Thus, \(14=70+(n-1)(-8)\) gives \(n-1=7\), so \(n=8\). Therefore, 14 is the 8th term. The 7th term is 22, so it is a close but incorrect option. Exam tip: use a negative common difference for a decreasing AP.

The common difference of an arithmetic progression is found by subtracting a term from the next term. Here, for every pair of consecutive terms, \(3-3=0\); therefore, the common difference is \(0\). The number \(3\) is the value of each term, not the common difference. Exam tip: subtract any term from its next term to check the common difference.

The first term is (19) and (d=4), so (a_n=19+(n-1)4=4n+15). Remember (n-1) while forming the general term.

From (a_4) to (a_9), there are (5) gaps, so (a_9=22+5(6)=52). Moving forward from a nearby term is a quick method.

The first term is (13), and adding (5) each time gives (13,18,23,28). Form the terms directly from the given (a) and (d).

In an arithmetic progression (2(x+6)=x+2x), so (x=12). The middle term is the average of the two surrounding terms.

(a_8=3) and (a_9=-3), so the first negative term is the (9)th. Check the first term that goes below zero.

For the third term, substitute n=3: \(a_3=11-2(3)=11-6=5\). Therefore, the correct answer is 5. Note that 7 is the second term, since \(a_2=11-2(2)=7\). Exam tip: In an nth-term formula, substitute the position number of the required term for n.

The average of the first three terms is (\frac{6+14+22}{3}=14). In an arithmetic progression the average of three consecutive terms is the middle term.

In an arithmetic progression, the nth-term formula is \(a_n=a_1+(n-1)d\). Thus, \(50=5+(10-1)d=5+9d\). Hence \(9d=45\), so \(d=5\). Option 4 is incorrect because there are 9 common-difference gaps from the first term to the tenth term, not 10. Exam tip: always use \((n-1)d\) in the nth-term formula.

In this arithmetic progression, the first term is \(a=2\) and the common difference is \(d=10-2=8\). The \(n\)th term is \(a_n=a+(n-1)d\). Therefore, \(a_{15}=2+(15-1)\times 8=2+112=114\). Hence, \(114\) is correct. \(118\) can result from incorrectly using \(15\times 8\); the formula uses \((n-1)\). Exam tip: identify the first term and common difference before substituting in the nth-term formula.

In (9,18,27,36,\ldots), (9) is added each time. To find the common difference check the difference of two consecutive terms.

From (6n+1=49), (6n=48) and (n=8). To find the position set (a_n) equal to the given value.

The first (5) terms are (30,25,20,15,10), and their sum is (100). In a decreasing progression also list the terms in order and add.

From (a_6) to (a_3), move back (3) gaps, so (a_3=41-3(7)=20). When moving backward subtract the common difference.

The first term is \(a=24\) and the common difference is \(d=-6\). Using \(a_n=a+(n-1)d\), we get \(0=24+(n-1)(-6)\), so \(n=5\). Hence, 0 is the fifth term. The fourth term is 6, so it is not correct. Exam tip: To find a term’s position, substitute the given value for \(a_n\) and solve for \(n\).

The nth term of an arithmetic progression is \(a_n=a+(n-1)d\). Substituting \(a=17\), \(d=0\), and \(n=10\) gives \(a_{10}=17+9\times0=17\). Hence, every term is 17. The value 170 comes from multiplying the first term by 10, which is not the rule for finding the nth term. Exam tip: when \(d=0\), every term of an AP equals its first term.

There are (6) gaps between (a_{11}) and (a_5), and (d=6), so the difference is (36). In an arithmetic progression term difference depends on the difference of positions.

The rule is \(a_n=10-3n\). Substituting \(n=1,2,3,4\) gives \(a_1=7\), \(a_2=4\), \(a_3=1\), and \(a_4=-2\), respectively. Hence, the required terms are \((7, 4, 1, -2)\). Option B, \((10,7,4,1)\), would result if counting started from \(n=0\), whereas the first term is normally obtained using \(n=1\). Exam tip: always check the starting value of the index before listing terms.

The first (6) terms are (10,16,22,28,34,40), and their sum is (150). Write the terms in correct order and add.