Question

1. 12, 112, 212, 312,...
2. -32, -52, -72, -92,...

find the common difference, the 52nd term, and the explicit formula. $a_n=a_1 + d(n - 1)$

1. -19, -119, -219, -319,...
2. -37, -43, -49, -55,...
3. 0, 30, 60, 90,...
4. -31, -40, -49, -58,...

Explanation:

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Step1: Find common difference

$d = 112 - 12=100$

Step2: Find explicit formula

Using $a_n=a_1 + d(n - 1)$, with $a_1 = 12$ and $d = 100$, we get $a_n=12+100(n - 1)=100n - 88$

Step3: Find 52nd term

Substitute $n = 52$ into the formula: $a_{52}=12+100\times(52 - 1)=12+100\times51 = 5112$

Step1: Find common difference

$d=-52-(-32)=-20$

Step2: Find explicit formula

Using $a_n=a_1 + d(n - 1)$, with $a_1=-32$ and $d = - 20$, we get $a_n=-32-20(n - 1)=-20n - 12$

Step3: Find 52nd term

Substitute $n = 52$ into the formula: $a_{52}=-32-20\times(52 - 1)=-32-20\times51=-32 - 1020=-1052$

Step1: Find common difference

$d=-119-(-19)=-100$

Step2: Find explicit formula

Using $a_n=a_1 + d(n - 1)$, with $a_1=-19$ and $d=-100$, we get $a_n=-19-100(n - 1)=-100n + 81$

Step3: Find 52nd term

Substitute $n = 52$ into the formula: $a_{52}=-19-100\times(52 - 1)=-19-100\times51=-19 - 5100=-5119$

Answer:

Common difference: $100$
52nd term: $5112$
Explicit formula: $a_n = 100n - 88$

26)