The sum of the first 30 terms of an arithmetic progression is 930. If the first term is 2, what is the common difference of the progression?
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Use the formula for the sum of an arithmetic progression to solve for the common difference when the sum and first term are known. Ensure the correct substitution of values in the equation.
The sum of the first 30 terms of an arithmetic progression (AP) is 930, with the first term being 2. To find the common difference \( d \), we use the AP sum formula: \[ S_n = \frac{n}{2} \left[ 2a + (n-1) \cdot d \right] \] Given \( S_{30} = 930 \), \( a = 2 \), and \( n = 30 \): \[ 930 = \frac{30}{2} \left[ 2(2) + (30-1) \cdot d \right] \] \[ 930 = 15 \left[ 4 + 29d \right] \] \[ 930 = 60 + 435d \] \[ 930 - 60 = 435d \] \[ 870 = 435d \] \[ d = \frac{870}{435} \] \[ d = 2 \] The common difference is \( d = 2 \).