15. Find the number of terms in the AP series 2,8,32,128,ldots 50 A. 11 B. 13 C. 15 D. 16

To find the number of terms in the given arithmetic progression (AP) series, we can use the formula for the nth term of an AP:<br /><br />\[ a_n = a + (n-1)d \]<br /><br />where:<br />- \( a \) is the first term,<br />- \( d \) is the common difference,<br />- \( n \) is the number of terms.<br /><br />Given the series \( 2, 8, 32, 128, \ldots \), we can observe that the common difference \( d \) is not constant. This suggests that the series might not be a simple arithmetic progression. Let's re-examine the series to see if it follows a different pattern.<br /><br />The series given is:<br />\[ 2, 8, 32, 128, \ldots \]<br /><br />We can see that each term is multiplied by 4 to get the next term:<br />\[ 2 \times 4 = 8 \]<br />\[ 8 \times 4 = 32 \]<br />\[ 32 \times 4 = 128 \]<br /><br />This suggests that the series is actually a geometric progression (GP) with the first term \( a = 2 \) and the common ratio \( r = 4 \).<br /><br />The nth term of a geometric progression is given by:<br />\[ a_n = ar^{n-1} \]<br /><br />We need to find the number of terms in the series up to 50. So, we set \( a_n = 50 \) and solve for \( n \):<br /><br />\[ 50 = 2 \times 4^{n-1} \]<br /><br />Divide both sides by 2:<br /><br />\[ 25 = 4^{n-1} \]<br /><br />Take the logarithm of both sides (base 4):<br /><br />\[ \log_4(25) = n - 1 \]<br /><br />\[ n - 1 = \log_4(25) \]<br /><br />Since \( 25 = 5^2 \), we can use the change of base formula to find \( \log_4(25) \):<br /><br />\[ \log_4(25) = \frac{\log(25)}{\log(4)} \]<br /><br />Using the natural logarithm:<br /><br />\[ \log_4(25) = \frac{\ln(25)}{\ln(4)} \]<br /><br />Calculate the values:<br /><br />\[ \ln(25) \approx 3.218 \]<br />\[ \ln(4) \approx 1.386 \]<br /><br />\[ \log_4(25) \approx \frac{3.218}{1.386} \approx 2.32 \]<br /><br />So,<br /><br />\[ n - 1 \approx 2.32 \]<br /><br />\[ n \approx 3.32 \]<br /><br />Since \( n \) must be an integer, we round up to the next whole number:<br /><br />\[ n = 4 \]<br /><br />Therefore, the number of terms in the series up to 50 is:<br /><br />\[ \boxed{4} \]