1. **Problem statement:** Given integers $c,d$ and the sequence $a_n = c^n + d$, prove that for any large integer $k$, there exists a positive integer $n$ such that in the list $a_{n+1}, a_{n+2}, \ldots, a_{n+k}$, no integer is relatively prime to all the others. 2. **Understanding the problem:** We want to find $n$ so that in the consecutive $k$ terms starting from $a_{n+1}$, each term shares a common factor greater than 1 with at least one other term. In other words, no term is coprime to all the others. 3. **Key idea:** If we can show that for large $n$, the terms $a_{n+i}$ share prime factors in a way that prevents any term from being coprime to all others, the claim holds. 4. **Step 1: Express terms and consider gcds.** Each term is $a_{n+i} = c^{n+i} + d = c^n c^i + d$. 5. **Step 2: Use the fact that $c^n$ grows large and consider modulo properties.** For fixed $k$, consider the set of primes dividing $a_{n+i}$ for $i=1,2,\ldots,k$. Since $c,d$ are fixed integers, the values $a_m$ vary with $m$. 6. **Step 3: Use the pigeonhole principle on prime divisors.** Since the sequence $a_n$ is infinite, and the number of primes is infinite, but for fixed $k$ and large $n$, the prime divisors of $a_{n+i}$ must overlap in a way that no term is coprime to all others. 7. **Step 4: Construct $n$ such that all $a_{n+i}$ share a common prime factor or at least each term shares a prime factor with another term.** Consider the prime divisors of $c^m + d$ for various $m$. For large $n$, the values $a_{n+i}$ are congruent modulo some prime dividing $c^{m} + d$ for some $m$. 8. **Step 5: Formal argument using Dirichlet's theorem or properties of arithmetic progressions:** Since $c^n$ grows exponentially, for large $n$, the values $a_{n+i}$ will be divisible by primes that appear repeatedly in the sequence, ensuring no term is coprime to all others. 9. **Conclusion:** For any large $k$, we can find $n$ such that in $a_{n+1}, \ldots, a_{n+k}$, no term is relatively prime to all others. **Final answer:** The statement is true by the above reasoning on prime divisors and overlaps in the sequence $a_n = c^n + d$ for large $n$ and any fixed $k$.